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Professor ALBERT EINSTEIN
READABLE
RELATIVITY
A BOOK FOR NONSPECIALISTS
BY
CLEMENT V. DURELL, JVLA.
SEMlOli MATHEMATICAL MASTER AT WINCllBSTER COULEGIS
When the ideas involved in Einstein's work have
become familiar, as they will do when they are
taught in schools, Certain changes in our habits
of thought are hkely to result, and to have great
importance in the long run."
Berth and Russell,
The ABC of Relativity.
LONDON
G. BELL & SONS LTD,
1926
TO
THE MEMBERS OF SENIOR BLOCK, DIVISION G
CLOISTER TIME, 1 924
WHO HELPED TO WRITE THIS BOOK
PRINTED IN GREAT BRITAIN BY MORRISON AND GIBB LTD., EDINBURGH
PREFACE
Relativity without mathematics may be compared with
" Painless Dentistry," or " Skiing without falling," or
" Reading without Tears." Its ideas have, of course, been
sketched in popular style by many writers, but precision
can only be achieved by setting out the arguments in a
mathematical form, and this precision is essential for a firm
grasp of the fundamental principles of the subject. This
book attempts to secure as high a degree of definition as is
compatible with the standard of mathematical knowledge
of the average person. The limitations this imposes are
obvious, but inevitable if the subject is to lie within the
sphere of general education. Given, however, this small
amount of mathematical capacity and preferably also a
willingness to work out a few numerical examples to test
appreciation of the ideas peculiar to the subject, it should
be possible to make Einstein's view of the Universe as
much a part of the intellectual equipment of ordinary
people as is that of Newton.
It is difficult to specify all the books from which I have
received assistance, but mention must be made of Pro
fessor Nunn's Relativity and Gravitation, and Professor
Eddington's Space, Time, and Gravitation and Mathematical
Theory of Relativity ; I am also indebted to Mr. Greenstreet
for allowing me to make use of the articles in the Mathe
matical Gazette to which reference is made in Chapter II,
vi PREFACE
and to Professor Eddington for information about the solar
eclipse expedition of 1922. The subjectmatter of the brief
note on Mr. Adam's work on the spectrum of the companion
star of Sirius is taken from Mr. Adam's published paper on
the subject, which Professor Turner very kindly showed me.
I have also to thank Mr. W. L. F. Browne for reading the
text and making a number of valuable suggestions.
C. V. D.
February 1926.
CONTENTS
CHAP.
I. The Progress of Science.
II. Alice through the LookingGlass
III. The Velocity of Light
IV. Clocks .
V. Algebraic Relations
VI. Separation of Events
VII. The Fourth Dimension
VIII. Mass and Momentum
IX. General Relativity
X. The Einstein Tests
Answers
PAGE
I
II
23
41
53
67
81
99
117
*33
145
READABLE RELATIVITY
CHAPTER I
THE PROGRESS OF SCIENCE
" We know very little and yet it is astonishing that we know
so much, and still more astonishing that so little knowledge
can give us so much power." — Bertrand Russell, The
ABC of Relativity.
Our Outdoor World.
Every one nowadays has heard of Einstein and Relativity.
The man in the street, however, still treats the subject
more as a fairy tale or a mathematical jest than as a
contribution to scientific knowledge and method which is
as momentous (if it stands th« test of continued scrutiny)
as any that has ever been made. Relativity is a branch of
physics, not of pure mathematics. Its conclusions could
not, of course, have been obtained without the aid of pure
mathematical reasoning of a difficult and abstruse nature,
but the mathematical side is incidental : mathematics has
merely supplied the machinery for working on the material
available and the language for describing the result. The
material itself i& the product of experiment, observation, and
measurement. The whole aim of Natural Science is to
examine what is happening in our outdoor world, the
Universe in which we live, and to construct the simplest
set of suppositions which will cover all the observed facts.
The sole question, therefore, that arises is this : Does
2 READABLE RELATIVITY
Einstein's theory of Relativity give a more harmonious
and adequate picture of what is observed to be happening
in the Universe than any other alternative scientific theory
dealing with the same data, or not ?
Many articles in newspapers and magazines seem to
suggest that the Relativity creed contains such fantastic
propositions that, however much they may appeal to
philosophers, they cannot be taken seriously by the matter
offact individual who prides himself on believing only
what he sees and distrusts suggestions which seem to
contradict his own personal experience. It is therefore
essential to realise that Einstein's theory stands or falls by
observed facts : it aims at describing how things such as
matter, time, and space do really behave. If, and when,
any one can offer a simpler and more comprehensive account
of the outdoor world than Einstein's theory is able to do,
then Relativity will be superseded or modified. But at the
present time it is contended that no other picture of the
Universe, as adequate as Einstein's, exists. The story of
scientific progress illustrates how indigestible new ideas are
as a diet for plain, blunt men, but equally that those ideas
which stand the test of time are easily assimilated by the
plain, blunt men of a later date.
The Ptolemaic System.
Tradition suggests that Pythagoras (550 B.C.) was the
first to teach that the Earth is a sphere, poised in space.
This puzzled the man in the street in two respects : (i) How
was the Earth supported ? (ii) Why did not people or
things on the other side of the Earth, being upside down,
fall off ?
Here, then, was one of the first of a series of shocks
scientists have administered to common sense and will
doubtless continue to administer as long as knowledge
increases. But many centuries passed before the sphericity
THE PROGRESS OF SCIENCE 3
of the Earth was generally accepted, at any rate outside
Greece ; in fact it needed the circumnavigation of the
world to drive it home as a matter of practical experience.
Even today there are some who still maintain the Earth
is flat (cf. Kipling's story, " The Village that voted the
Earth was Flat," in A Diversity of Creatures), just as there
are others who still try to square the circle. The Greek
scientists, however, waived criticism aside and proceeded
to calculate the circumference and diameter of the Earth.
Aristotle (350 B.C.) states that the mathematicians of his
time made the circumference 400,000 stadia (probably
about 40,000 miles), but a far more accurate result was
obtained by Eratosthenes (250 B.C.), who gave the circum
ference as the equivalent of 24,700 miles and the radius
as 3925 miles, a degree of accuracy far higher than the
rough methods he employed justify ; it was in fact a
lucky result. A later but less accurate measurement was
made by Ptolemy (140 a.d.), whose treatise, largely astro
nomical, The Almagest, dominated scientific thought for
the next fourteen centuries. The Ptolemaic System placed
the Earth at the centre of the Universe, and regarded the
Sun and the planets as moving round it in paths built up
of circle and epicycles (i.e. circles rolling on circles) ; and
although as time passed minor modifications were made
to bring the theory into closer accord with observation,
the general principle was accepted without question, till
the time of Copernicus. It may be of interest to record
that a contrary view had been put forward before Ptolemy's
time. A certain Aristarchus of Samos (310230 B.C.)
maintained that the Sun was the centre of the world and
that both the planets and the Earth moved round it. But
this was far too advanced a doctrine not only for the
common man but also for the scientists of his day ; it
offended their sense of propriety in that it degraded the
Earth from the central to a subordinate position in Nature,
4 READABLE RELATIVITY
and it outraged common sense to suppose that the Earth,
which, as they said, any one could see and feel to be at
rest, was really travelling through space more than a
million miles a day. [Eratosthenes estimated the distance
of the sun as 80,000,000 miles.] We are all now so accus
tomed to think of the Sun as the centre of the solar
system that it is difficult to realise what a shock the
normal man received when the doctrine was first seriously
propounded.
The Copernican System.
The theory of Copernicus, De Revolutionibus orbium
Coelestium, was published in 1543. It regarded the Sun as
at rest at the centre of the world with the Earth and the
other planets moving round it in circular orbits. The
laborious work of Kepler (15711630), although showing
that the orbits were not circular, gave powerful support
to the heliocentric principle of Copernicus by establishing
that the planets could be regarded as moving in ellipses
with the Sun at one focus, and that the sizes of the orbits
and the times and rates of description obeyed two simple
quantitative laws, which later on assisted Newton to
formulate his law of Universal Gravitation. The invention
of the telescope in 1608 led Galileo (15641642) to discover
Jupiter's moons, and this smallscale model of the solar
system convinced him of the truth of the Copernican
theory. After much hesitation and with considerable
trepidation, which afterevents fully justified, Galileo pub
lished in 1630 an account of his discoveries and beliefs.
His thesis was not only a shock to the man in the street
but, unhappily for Galileo, a shock also to the Church.
In selfdefence, he begged his opponents to come and look
through his telescope, but neither the professors nor the
ecclesiastics would do so. He was summoned to Rome,
an old man in feeble health, tried by the Inquisition, and
THE PROGRESS OF SCIENCE 5
forced to " abjure, curse, and detest his errors and heresies."
He died, a broken man and blind, near Florence in 1642.
Einstein has good cause to be thankful that there is no
Inquisition in power today.
Universal Gravitation.
The work of Copernicus, Kepler, Simon Marius, and
Galileo was crowned by the publication of Newton's
Principia in 1685. The notion that bodies fell to the earth
owing to some form of attraction exerted by the earth did
not originate with Newton. His genius showed itself in
extending this idea to the whole Universe, formulating his
result in a single law, and verifying it by an examination
of the motion of the planets, comets, the Earth, and the
Moon. Newton's troubles lay with his brotherscientists,
not with ordinary folk, and many years of his life were
embittered by the professional controversies which the
Principia evoked. It has been said that there are com
paratively few scientists who can, at the present time,
really understand the mathematical work which is the
scaffolding of Einstein's theory, but there were relatively
far fewer in Newton's day who could appreciate the
reasoning of the Principia; and of course a long time
elapsed before Newton's ideas became part of the equip
ment of the ordinary educated man, as they are today.
The Mechanics of Galileo and Newton.
It is necessary to state, however briefly, the fundamental
principles upon which Newton, using the observations of
Kepler and the ideas of Galileo, based his system of celestial
mechanics. He followed Galileo in saying that the nature
of a body is such that, if at rest it will remain at rest, and
if in motion it will continue to move uniformly in a straight
line, unless there is some external cause operating to pro
duce a change. This is the Principle of Inertia, and to
6 READABLE RELATIVITY
the external cause he gives the name of Force and there
from develops his idea of Mass. 1
For the working out of his mechanics, he postulates :
(i) The notion of absolute time : time flows uniformly
and without reference to anything else.
(ii) The notion of absolute space : a fixed standard
of reference, immutable and immovable, enabling the
position or motion of any object in the Universe to be
determined. The Earth is not at rest, the Sun may not
be at rest, but there is, so Newton says, something existing
in the Universe which will act as a fixed frame for defining
absolute position and absolute motion.
The normal man neither had nor has any difficulty in
assenting to these suppositions ; in fact they seem so
natural that he is shocked when asked to question them.
The idea of universal gravitation was, on the other hand,
far more perplexing. It appeared to involve the notion
of " action at a distance," whereas everyday experience
pointed to the belief that the action of one body on another
was either caused by direct contact or some concrete
connecting agency. Newton himself appears to have
thought that further explanation was needed. Today it
is an idea that the normal man accepts without protest ;
for him time and tradition have, as always, acted as the
necessary shockabsorbers ; but none the less since Newton's
time a succession of scientists has attempted by a variety
of physical theories to bridge the gulf.
Measuring Instruments.
The human senses, unaided by mechanical assistance,
are not adapted for making accurate measurements either
of time or space.
Time. — Our sense of duration for periods of any length
depends mainly on whether our occupation is interesting
1 Fawdry, Readable Mechanics, ch. vii.
THE PROGRESS OF SCIENCE 7
or tedious : our estimate of short periods of time is often
ludicrously inaccurate. This may be illustrated by testing
the ability of any one to judg'e the length of a minute,
ruling out, of course, the assistance that counting, either
aloud or mentally, affords.
The use of the sundial, which treats the Sun as a clock,
is known to date back to 1500 B.C. in Egypt, and according
to tradition it was introduced into Greece from Babylon
by Anaximander in the sixth century B.C. ; about this time
also hourglasses and waterclocks began to be made. Clocks
depending on trains of wheels driven by a falling weight
were used by the Romans in the sixth century a.d. The
pendulum clock was invented by Huygens in 1673, ninety
years after Galileo had discovered the isochronous property
of a pendulum : Galileo himself used for his experimental
work a form of waterclock l which gave surprisingly good
results. The first ship's chronometer of a reliable nature
was constructed by a Yorkshireman, John Harrison, in
1761. This was an invention of immense practical import
ance, as at that time ships at sea could only determine
their longitude by means of a clock. Under present condi
tions it is of less importance, because Greenwich mean
time is sent out by wireless at noon each day.
Space. — Any one who visits Oxford should go and see
the Evans Collection of ancient scientific apparatus at the
old Ashmolean. The delicacy and accuracy of workman
ship of the rulers, compasses, and astrolabes {i.e. instru
ments for measuring angles fixing the positions of the
stars), some of which date back to very early times, can
only be appreciated by seeing them. Chaucer wrote a
treatise on the use of the astrolabe to make sure that his
son should be properly instructed in its theory when he
went to Oxford. The principle of the vernier, applied
first to circular arcs, was discovered by a Portuguese
1 Fawdry, Readable Mechanics, p. 62.
8 READABLE RELATIVITY
named Nunes in 1542, and rediscovered by the Frenchman
Vernier in 1631.
Artificial Extension of the Senses.
No great advance was possible in astronomy until the
invention of the telescope in 1608. Naturally such an
instrument at once aroused widespread attention, and
within a few years telescopes were being used by scientists
all over Europe. It not only increased the range of
observation in astronomy, but it raised the degree of
accuracy of measurement to a level far higher than had
previously been possible. Modern discoveries are largely
the fruit of two other inventions : (i) dryplate photography,
and especially its application to astronomy; (ii) the
spectroscope and spectrum analysis. It is outside the
province of this book to give any account of the amazing
variety of application and the remarkable refinements
of measurement yielded by these methods of research.
The important point to observe is that the increase of
knowledge of the Universe is due entirely to the successive
aids that inventions have given in supplementing the
powers of naked eyes and naked hands. Without such
assistance, our knowledge of the structure of the world
would remain very restricted. Without a telescope, we
can only see details of things in our immediate neighbour
hood, and that only to a limited extent ; to fill in further
detail, a microscope is necessary. Our ears only enable us
to hear notes of limited pitch ; our eyes are sensitive to a
range of colour far more limited than the range of the
photographic plate ; if events happen too rapidly, our
brains receive merely a blurred impression which it needs
a slowmotion film to disentangle.
Common Sense.
Most of our outlook on life is coloured by the impressions
we receive through senses, unaided by any artificial assist
THE PROGRESS OF SCIENCE 9
ance. What we call a commonsense view of life is largely
based on an acquaintance with things, confined in size
between fairly narrow limits, restricted to small ranges
of temperature and pressure, moving at low speeds and
for short periods of time. It is not unreasonable to say
that this gives us as true a view of the Universe as (say)
a tourist could obtain of the interior of Westminster
Abbey by looking through the keyhole of a side entrance.
Successive inventions have enabled scientists to enlarge
the keyhole, and perhaps at some future date will even
throw open the door. If Science by its study of things
very small and very large, very near and very distant, of
temperatures very great, of velocities very high, is driven
to conclusions which seem to violate our commonsense
attitude, our keyhole notion of the Universe, it seems
reasonable to treat it merely as one more shock in the
succession which the man in the street has encountered
and eventually absorbed. Nature is a conjurer for super
men. Generations of scientists have attempted to pene
trate her secrets. Bit by bit the disguise is being torn
away, but each new discovery seems only to open out
fresh avenues demanding further exploration. Nature is
a true woman, who will have the last word. Scientists
of every age may well echo Newton's account of his own
life's work : " I do not know what I may appear to the
world, but to myself I seem to have been only like a boy
playing on the seashore, and diverting myself in now and
then finding a smoother pebble or a prettier shell than
ordinary, whilst the great ocean of truth lay all undis
covered before me."
EXERCISE I
1. Why do you sometimes see on a cinematograph the wheels
of a car rotating in a clockwise direction while the car
itself is moving to the left ?
io READABLE RELATIVITY
2. A stone is tied to the end of a string and is whirled round
in a circle horizontally. In what direction does the
stone move when the string breaks ?
3. If you assume that the earth describes a circle of radius
93,000,000 miles in 365J days, how many miles does it
move in a second ?
4. Galileo found it easier to show that the acceleration of a
falling body is constant by considering motion down a
slope instead of vertically. How could a cinematograph
operator improve on Galileo's treatment ?
5. If you drop a stone and a feather at the same moment, do
they hit the ground simultaneously ? Is your answer
consistent with the statement that the acceleration due
to gravity is the same for all bodies at the same place,
regardless of their weights ?
6. A camera photographs a 100 yards race for a cinema film,
taking 150 exposures per second. They are reproduced
on the screen at the rate of 15 per second ; how long,
roughly, will the race appear to the audience to last ?
7. Describe a vernier, and show how one can be made to read
correct to yfoyth inch.
8. Taking the length of the Equator as 25,000 miles, find the
error in miles of the longitude of a position on the
Equator, calculated from the record of a chronometer,
if there is an error of one minute in the time.
9. Eratosthenes found that the sun was in the zenith at Syene
when it was 7 12' south of the zenith at Alexandria,
which was known to be 5000 stadia north of Syene.
What expression for the radius of the Earth can be
deduced from these data ?
CHAPTER II
ALICE THROUGH THE LOOKINGGLASS
" ' I can't believe that,' said Alice.
" ' Can't you ? ' the Queen said in a pitying tone. ' Try
again : draw a long breath and shut your eyes.'
" Alice laughed : ' There's no use trying,' she said ; ' one
can't believe impossible things.'
" ' I daresay you haven't had much practice,' said the
Queen. ' When I was younger, I always did it for half an
hour a day. Why, sometimes I've believed as many as six im
possible things before breakfast.' " — Through the LookingGlass .
Can Nature deceive ?
The scientists, in playing their game with Nature, are
meeting an opponent on her own ground, who has not only
made the rules of the game to suit herself, but may have
even queered the pitch or cast a spell over the visiting team.
If space possesses properties which distort our vision,
deform our measuringrods, and tamper with our clocks, is
there any means of detecting the fact ? Can we feel hope
ful that "eventually crossexamination will break through
the disguise ? Professor Garnett, 1 making use of Lewis
Carroll's ideas, has given a most instructive illustration of
a way in which Nature could mislead us, seemingly without
any risk of exposure.
Ultimately, we can only rely on the evidence of our
senses, checked and clarified of course by artificial appara
tus, repeated experiment, and exhaustive inquiry. Observa
tions can often be interpreted unwisely, as an anecdote told
by Sir George Greenhill illustrates :
1 Mathematical Gazette, May 191 8.
12 READABLE RELATIVITY
At the end of a session at the Engineering College,
Coopers' Hill, a reception was held and the science depart
ments were on view. A young lady, entering the physical
laboratory and seeing an inverted image of herself in a
large concave mirror, naively remarked to her companion :
" They have hung that lookingglass upside down." Had
the lady advanced past the focus of the mirror, she would
have seen that the workmen were not to blame. If
Nature deceived her, it was at least a deception which
further experiment would have unmasked.
The Convex LookingGlass.
We shall now follow some of the adventures of Alice in
a convex lookingglass world, as described by Professor
Garnett. As a preliminary, it is necessary to enumerate
some of the properties of reflection in a convex mirror. For
the sake of any reader who wishes to see how they can be
obtained, their proofs, which involve only the use of similar
triangles and some elementary algebra, are indicated in
Exercise IIb., No. i, p. 20.
The world of Alicia / The world of Alice behind
and Euclid j the lookingglass
P'
\h
y
Q' * F f
Fig. 1.
A is the apex of a convex mirror, of large radius ; O is its
centre, OA is the central radius or axis ; the midpoint F
of OA is the focus. PQ is an object outside the mirror,
perpendicular to the axis and of height a feet, P'Q' is the
image of PQ in the mirror. Denote the various lengths as
follows, in feet ;
ALICE THROUGH THE LOOKINGGLASS 13
OF=FA=/; FQ'=*; Q'A=* ; AQ=y ; V'Q'=h.
We have the following formulae :
1 1 . 1 7 ax .
* y f f
The general consequences of these formulae are easy to
appreciate.
Since =+y.we see that > , so that y>z.
and that 7 <, so that z<f or 2<AF.
The image P'Q' is therefore always nearer the mirror than
the object PQ is, and is never as far from the mirror as F is.
Again, since h=Y> tne height of the image is propor
tional to x, its distance from F, and therefore the nearer
P'Q' approaches F the smaller the length of P'Q', the height
of the image, becomes.
Life behind the LookingGlass.
" ' He's dreaming now,' said Tweedledee : ' and what do
you think he's dreaming about ? '
" ' Nobody can guess that,' said Alice.
" ' Why, about you ! ' Tweedledee exclaimed. ' And if he
left off dreaming about you, where do you suppose you'd be ? '
" ' Where I am now, of course,' said Alice.
" ' Not you ! ' Tweedledee retorted contemptuously.
' You'd be nowhere. Why, you're only a sort of thing in his
dream ! '
" ' If that there King was to wake,' added Tweedledum,
' you'd go out — bang ! — just like a candle ! '
" ' lam real ! ' said Alice, and began to cry.
" ' You won't make yourself a bit realer by crying,' Tweedle
dee remarked."
We shall now treat Alice not as a thing in a dream, but
as the image in a convex lookingglass of a pseudoAlice
who is moving about in our own world. Alice will insist
14 READABLE RELATIVITY
as vehemently as she did to Tweedledee that she is a free
agent with an independent existence, but we, looking
from outside, will see that she conforms to the movements
and amusements of this pseudoAlice, whom we will call
Alicia. We proceed to compare our (or Alicia's) observa
tions with Alice's own ideas about her mode of life.
Alice's Life.
Alicia is 4 feet tall and 1 foot broad. She starts at A
with her back against the mirror, so that she and Alice are
back to back, exactly the same size. Alicia is carrying a
footrule which she holds against the mirror so that it
touches and coincides with, and therefore equals, the corre
sponding footrule which Alice has.
Alicia now walks at a steady rate of 1 foot per second
away from the mirror, along the axis. What happens to
Alice ?
Suppose the radius of the mirror is 40 feet, so that
AF=F0=/=20 feet. Then a =PQ= Alicia's height=4 ft.
After (say) 5 seconds, AQ=v=5.
Then+ 7 g+ ?
/. AQ'=z=4 and x=Q'F=f— z=20— 4=16.
™™ , «* 4x16 32
.*. P'Q'=^=7= 1 ^=fe=32 feet.
If at this moment Alicia looks round, she will notice that
Alice has only moved 4 feet compared with her own 5 feet,
and that Alice's height has shrunk to 32 feet.
Alice's footrule, held vertically, has also shrunk: 'its
1 *u • x 2. ■ 1 X * 1 X 16
length m fact is now —f~= „ =o8 foot.
J 20
Alice repudiates the idea that she has grown smaller, and
to convince Alicia she takes her footrule and uses it to
measure herself, and shows triumphantly that she is still
exactly four footrules high (32^08=4).
ALICE THROUGH THE LOOKINGGLASS 15
Alicia also notices that Alice is not so broad as she was ;
her breadth, in fact, has now dwindled to o8 foot : true,
the breadth is still equal to Alice's footrule, but that foot
rule in any position perpendicular to the axis is now only
o8 foot long.
(For further numerical examples, see Exercise IIa., Nos. 13.)
ContractionRatio Perpendicular to the Axis.
We see that Alice continues to contract as she moves
farther away from the mirror. The contractionratio in
. h x
any direction perpendicular to the axis is  which equals j
and is therefore proportional to x, Alice's distance from the
focus F.
It is evident that Alice cannot detect this contraction,
because Alice's ruler contracts in just the same proportion
as Alice's body and Alice's clothes. In fact everything in
Alice's world, regardless of what it is made, behaves in
exactly the same way. We therefore call this contraction
a property of space, not a property of matter. It is a
form of influence which the space exercises on all things
alike which enter it. And we say that one of the laws of
space in Alice's world is an automatic contractionratio which
for any direction perpendicular to the axis, is proportional to
x, the distance from the focus.
ContractionRatio along the Axis.
Alicia now lays her footrule down along the axis, and of
course Alice imitates her.
1 1 — £ 1 W+
S Q A Q»S
Fig. 2.
Q, S are two successive points of division on Alicia's
footrule, such that AQ=5 feet, QS=oi foot, so that
i6
READABLE RELATIVITY
AS=5i feet. The corresponding marks on Alice's foot
rule are Q', S'. We have already proved that if AQ=v=5,
then AQ'=z=4. Further, if AS=v=5i, then AS' =2 is
given by
j_I + I_ I.I _ 2Q+5I ^25I
' z y f 5*1 20 51X20 102'
. _, 102
.". Ab =z=—— =4064, approximately.
/. Q'S'=AS'AQ' =0064 foot.
.*. the contractionratio along the axis at Q' is
Q'S' = oo64 r
QS oi
But the contractionratio perpendicular to the axis at Q'
has been shown to be o8.
Since (o8) 2 =o64, this suggests that the contraction
ratio along the axis is the square of the contractionratio
perpendicular to the axis, at the same place.
=064.
Fig. 3.
The proof of this statement is indicated in Exercise IIb.,
No. 2. Alicia therefore notices that Alice becomes thinner
ALICE THROUGH THE LOOKINGGLASS 17
from front to back as she moves away from the mirror,
and the rate of getting thinner is more rapid than the rate
of getting shorter.
For example, if Alice turns sideways and stretches
out her left arm towards the mirror and her right arm
away from it, the fingers of her left hand will be longer
and fatter than those of her right hand, but the general
effect will be to make the fingers of her right hand appear
puffier — a chilblain effect — because these fingers have
shortened more than they have thinned, compared with the
other hand.
EXERCISE IIa
[In this exercise, Alicia is supposed to be 4 feet high and her
waist measurement is 1 foot broad, 6 inches thick; also /= 20 ft.]
1. When Alicia has moved 10 feet from the mirror, show that
Alice has only moved 6 feet 8 inches, and is now 2 feet
8 inches tall and 8 inches broad. What happens when
. Alice measures her height with her own footrule ?
2. When Alicia has moved 20 feet from the mirror, show that
Alice has only moved 10 feet. What is Alice's height
and breadth in this position ?
3. Where is Alicia, when Alice's height is reduced to 1 foot ?
What is then Alice's breadth ? What is the length of
her footrule, held vertically ? How many footrules
high is Alice ?
4. With the data of No. 1, find the thickness of Alice's waist.
What is Alice's measurement of it ?
5. With the data of No. 2, what is the contractionratio for
Alice along the axis ? What is the connection between
the contractionratios along and perpendicular to the
axis ?
6. Alicia is 20 feet from the mirror and holds a iinch cube
with its edges parallel or perpendicular to the axis.
What is Alice holding ?
7. When Alice's height shrinks to 1 foot, what are her waist
measurements ?
8. Does Alice's shape, as well as her actual size, alter as she
moves away from the mirror ? When Alice's height has
2
18 READABLE RELATIVITY
dwindled to i foot, a statue is made of her on an enlarged
linear scale, 4:1. Will this statue be a good lifesize
likeness of Alicia ?
Alice's Geometry.
While Alicia is walking away from the mirror at a
uniform speed, taking steps of equal length, Alice is also
walking away in the opposite direction ; but (according to
Alicia) Alice's steps get shorter and shorter, and so she
advances more and more slowly ; and in fact, however far
Alicia travels, Alice herself can never get as far as F.
Alice of course imagines that there is no limit to the
distance she can travel, and what Alicia calls the point F,
Alice calls a point at infinity. If Alice walks along level
ground, she imagines that the tip of her head and the soles
of her feet are moving along parallel lines ; indeed the
lines along which they move express Alice's idea of parallel
ism. Alicia sees that such lines actually meet at F.
Fig. 4.
If Alice lays a railway track along the axis, the railway
lines will behave in the same way.
Suppose Alicia is riding on a bicycle down the axis
away from the mirror, how do the wheels of Alice's
bicycle behave ?
The contraction is greater along the axis than at right
angles to it. Consequently, not only are the wheels of
ALICE THROUGH THE LOOKINGGLASS 19
Alice's machine smaller than those of Alicia's, but Alice's
front wheel is smaller than her back wheel : moreover, each
wheel is approximately elliptical in shape, its vertical
diameter being greater than its horizontal diameter, and
although the wheels are turning round, the spokes which
are vertical always appear to be longer than any of the
others, and the spokes which are horizontal always appear
to be shorter than any of the others ; the consequence is
that the spokes appear to expand as they revolve from the
horizontal to the vertical, and then to contract as they
revolve from the vertical to the horizontal.
Fig. 5.
Alice herself, after careful measurement, is satisfied that
the machine is quite normal, but Alicia will think it most
unsteady. Space does not permit of any inquiry into the
mechanics of Alice's life : for this, reference should be
made to Professor Garnett's article, mentioned above.
Innocents at Play.
The object of this chapter is not to suggest that we are
living in a lookingglass world, but to point out that there
would appear to be no method of discovering the fact, if
20
READABLE RELATIVITY
it were true. When Nature makes her laws of space, she
can cast a binding spell over its inhabitants, if she cares to
do so. But the fact that Nature is willing to answer some
of the experimental questions which scientists put en
courages them to think that gradually these laws of space
and time may be disclosed. The purpose of this chapter will
be served if it suggests that the search is not simple, and the
results may be surprising.
EXERCISE IIb
I. BAC is a convex mirror whose radius AO is large compared
with the object PQ. The ray of light from P parallel
Fig. 6.
to the axis meets the mirror at N, and is then reflected
along the line joining N to the focus F ; the ray of light
from P towards the centre O of the mirror is not changed
in direction by the mirror ; therefore the image of P is
at the intersection P' of NF and PO. Draw P'Q'
perpendicular to OA, then P'Q' is the image of PQ.
The focus F is at the midpoint of OA. Since the mirror
has a large radius, its curvature is small, and NA can be
treated as a straight line perpendicular to OA.
FO _FP'_FQ'
(i) Prove that
PN P'N Q'A
ALICE THROUGH THE LOOKINGGLASS 21
(ii) Hence, with the notation of p. 13, prove that
z y^f
P'O' O'F ax
(iii) Prove that ■jn=p and hence prove that h=~j.
(iv) Hence show that x=f—z=— and that = F .
V J J y y J
If in the figure PQ moves along the axis a short distance to
RS, and if the image of RS is R'S', and if AS=y 1 , AS>'=z 1 ,
use the formula = I + ^ to prove that
Q'S' ^— z _ zz ± ri _z 2 _% 2 /h\ 2
QS — y x — y~yy~ y z P W "
What do you deduce about the longitudinal contraction
ratio ? Those acquainted with the calculus should
}iz Sv
show that a=^, and interpret the result.
What does Alicia think of the movement of the hands of
Alicia's watch (i) when held facing the mirror (ii) when
laid flat on the ground ?
Alice spins a top so that its axis is vertical ; what is there
unusual about it, according to Alicia ?
The axis of the mirror A>0 points due east. Alice, whose
height has become only half that of Alicia, turns and
walks northeast. What is her direction as measured
by Alicia ?
Alice believes she has proved two given triangles congruent
by the method of superposition. Does Alicia agree with
her ?
CHAPTER III
THE VELOCITY OF LIGHT
" The first thing to realise about the ether is its absolute
continuity. A deepsea fish has probably no means of appre
hending the existence of water ; it is too uniformly immersed
in it : and that is our condition in regard to the ether." — Sir
Oliver Lodge, Ether and Reality.
The Ether.
Those who have engaged in physical research during
the last hundred years have been rewarded by discoveries
of farreaching importance and interest. Light, electricity,
magnetism, and matter have been linked together so
closely that it now appears that each must be interpreted
in terms of a single medium, the ether.
■;■ What the ether is will, no doubt, remain a subject of
acute controversy between physicists, of rival schools for
many years to come. Its existence was first postulated
to serve as a vehicle for light. Experiment showed that
light travels through space at approximately 300,000 km.
per second, thus light takes about 8 J minutes to reach the
Earth from the Sun.
According to the undulatory theory, light is propagated
in the form of a wave motion through the ether. The work
of Weber, Faraday, ClerkMaxwell, and others established
the remarkable fact that electromagnetic radiation is a
wave motion propagated with exactly the same velocity
as light, and therefore presumably it uses the same medium
as a vehicle. More recent research has shown that electric
24 READABLE RELATIVITY
charges are discontinuous, and that the atoms of which
matter is composed may themselves each be resolved into
a group of electric particles ; the group which constitutes
each atom contains both negatively charged particles
called electrons and positively charged particles called
protons ; atoms differ from each other according to the
number and grouping of these particles. Each atom may
be regarded as a miniature ultramicroscopic solar system,
in which the electrons describe orbits round a central
nucleus.
The ether is to be regarded as a continuous medium,
filling the whole of space, and may indeed be identified with
space : matter, electricity, etc., are discontinuous. But
while identifying ether with space, we must also attribute
to it some physical qualities in order that it may serve as
a vehicle for physical phenomena. Some physicists credit
it with weight and density. But if Einstein's view is
accepted, it has no mechanical properties of this nature.
Einstein holds that the idea of motion in connection with
the ether is meaningless; the ether is everywhere and
always. He does not say that the ether is at rest, but that
the property of rest or motion can no more be applied to
the ether than the property of mass can be applied to a
man's reflection in a mirror, although the lightrays by which
we perceive the reflection may and indeed do possess
mass.
Absolute Motion.
If a passenger in a train observes another train moving
past him, and if the motion is uniform and if there are no
landmarks in view, it is impossible for him to determine
whether his own or the other train, or both, are really in
motion. This is a familiar experience. There is no
difficulty in measuring the relative velocity of the two
trains, but without a glimpse of the ground to act as a
THE VELOCITY OF LIGHT 25
reference for measurement it is impossible to find what we
tend to call the actual velocity of the train.
Again, suppose two balloons are drifting past each
other above the clouds : an observer in one balloon tends
to think of himself at rest and the other balloon as moving
past him. Even when he obtains an accurate ground
observation, he can only calculate his velocity relative to
the earth. An astronomer might continue the work and
tell him the velocity of the observed point of the ground
relative to the Sun and then the velocity of the Sun relative
to one of the " fixed stars." But even all that will not enable
him to find his actual or absolute velocity. What reason is
there to consider any of the stars as fixed ? we know that
they also move relatively to each other ; what indeed can
the word " fixed " mean at all ? Is there anything in the
Universe we can mark down as really fixed ? Scientists
did not like the idea that all measurements of motion must
be relative; it seemed like building a structure of the
mechanics of the Universe on a shifting sand. When,
therefore, physical research demanded the existence of a
medium filling the whole of space, the ether was welcomed
not only for what it could do for light and electricity but
because it appeared to offer a standard of reference for the
measurement of absolute velocity. Scientists, therefore,
set to work to measure the velocity of the Earth through
the ether. The fundamental experiment which had this
object, and which may be taken as the basis for describing
Einstein's (restricted) theory of Relativity, was performed
in 1887 by Michelson and Morley. We shall in future
refer to it as the M.M. experiment. The idea of that experi
ment may be easily understood by taking a simple analogy.
Rowing on Running Water.
A stream is flowing at 4 feet per second between straight
parallel banks 90 feet apart. Two men start from a point
26 READABLE RELATIVITY
A on one bank ; one of them T rows straight across the stream
to the opposite bank at B and returns to A, the other L rows
to a point C 90 feet downstream and then rows back to A.
Each of them rows at 5 feet per second relatively to the water.
Compare their times.
We have called the oarsmen T and L because T rows
transversely to the
5 stream and L rows
go
©
©
longitudinally, in the
otjft./sec line of the stream.
~" Now T, in order to
reach B, must point
his boat upstream
along a line AP such
A 90 ' C that if AP=5feet (i.e.
Fig. 7. the distance he moves
relative to the water
in 1 second), the water will carry him down 4 feet from P
to Q (i.e. the distance the stream runs in 1 second), where
Q is a point on AB and ZAQP is a right angle.
By Pythagoras, AQ 2 +4 2 =5 2 , .'. AQ 2 =25— 16=9,
/. AQ= 3 feet.
Therefore in each second the boat makes 3 feet headway
along AB.
.". the boat takes ^=30 seconds to get from A to B.
Similarly it takes 30 seconds to return from B to A.
.*. the total time across and back =60 seconds.
Now L, on his journey to C, is moving relatively to the
water at 5 feet per second, and the water carries him for
wards 4 feet per second ; therefore he advances at the
rate of 5+4=9 ^ ee ^ P er second.
.*. the time from A to C=9=io seconds.
9
THE VELOCITY OF LIGHT 27
But when returning against the stream from C, his advance
is only 5—4=1 foot per second.
.'. the time from C to A =9°. =90 seconds.
.". the total time downstream and up=io+ 90=100 seconds.
/ # time of L down and up __ioo^5
• • time of T across and back 60 3"
It therefore takes longer to go down and up than an equal
distance across 'and back. But the working of this example
also shows that if we know the rate of rowing relative to
the water, and if we know the ratio of the times taken for
equal journeys in the two directions, we can calculate the
velocity of the stream.
The MichelsonMorley Experiment.
It is known by experiment that light always travels
through the ether at a constant rate of 300,000 km. per
second. Let us suppose that at a
certain moment the Earth is moving
through the ether at a speed of u km.
per second in the direction C>A, then
from the point of view of a man on
the Earth the ether is streaming past
A in the direction A»C at u km. per y\ q
second. AC and AB are two rigid Fig. 8.
equal and perpendicular rods, with
mirrors attached at C, B, so as to face A. At the same
moment rays of light are dispatched from A, one along
AC and the other along AB ; these rays impinge on the
mirrors and are reflected back to A. The motion of these
rays corresponds to the motion of the boats in the ex
ample given above. Each ray travels at 300,000 km. per
second relative to the etherstream, since its mode of
propagation is a wave in the ether, just as each boat
28 READABLE RELATIVITY
moves at 5 feet per second relative to the water. Also
the etherstream is itself moving in the direction A>C
at u km. per second, just as the waterstream is flowing
at 4 feet per second.
Now it takes longer to go any given distance downstream
and back than to go the same distance across the stream
and back. Consequently the ray from C should arrive
back at A later than the ray from B. If then we measure
the ratio of the times taken by the two rays, we can calculate
the speed u km. per second of the etherstream, and this
is equal and opposite to the velocity of the Earth through
the ether.
The M.M. experiment was designed to measure the
ratio of these times : for a detailed account of the apparatus
employed, reference may be made to any standard modern
textbook on light. To the astonishment of the experi
menters, the race proved to be a dead heat, the ray from
C arriving back at A simultaneously with the ray from B.
Now the earth is describing its orbit round the Sun at
a speed approximately of 30 km. per second, consequently
there is a difference of speed after a
six months' interval of about 60 km.
per second, so that even if the Earth
should happen to be at rest in the
FlG ether at one moment, it could not
be still at rest six months later. But
the repetition of the experiment after a six months'
interval still gave a dead heat.
Also, in order to guard against any error arising from
an inequality of the lengths of the arms AB and AC, the
experiment was repeated after rotating the arms so that
AB lay along the supposed stream and AC across it ; but
no difference of time was detected. Further, different
directions were tried for AB, but without any result. The
experiment has been carried out more recently with such
THE VELOCITY OF LIGHT 29
added refinements that as small a speed as ^ km. per second
would have been detected. Here, then, was an experimental
result which contradicted a conclusion obtained by theory.
Clearly there was something wrong with the theory.
Scientists were compelled to look for some explanation or
some modification of the theory which would reconcile
calculation with observation.
What is the Answer to the Riddle ?
Let us return to the illustration of the boats which
correspond to the lightrays in the M.M. experiment. The
two boats start together under the conditions stated on
p. 26, and every one is then amazed to see them arrive
back simultaneously. How is this to be reconciled with
the conclusions reached by calculation ?
The first suggestion is that L rowed faster relatively to
the water than T ; this, however, must be rejected because
the speed of rowing through the water
corresponds in the M.M. experiment to li
the velocity of light through the ether,
which we know is a constant, 300,000 km.
per second.
The next suggestion is that the courses
are marked out incorrectly, and that
the length of AC is less than that of AB, p IG IO>
owing to careless measurement. But this
view is untenable, because in the M.M. experiment when
the rigid arms AB'and AC were interchanged, there was
still no difference of time.
Fitzgerald then suggested that the arms AB and AC
were unequal, not through faulty measurement, but
because the shifting of a bar from a position across the
stream to a position along the stream caused automatically
a contraction in the length of the bar. The adventures of
Alice have shown us that such a contraction could not be
©
30 READABLE RELATIVITY
discovered by measurement, because the footrule with
which AC is measured contracts in just the same proportion
as the arm.
Suppose in the example of the boats the rule which
measures i foot across the stream contracts to § foot when
in the line of the stream. When we measure out 90 foot
rule lengths to obtain AC, the outsider (Alicia) will say
that AC is really f X 90 =54 feet, instead of 90 feet. L will
then take ^4=6 seconds to go downstream and 54
9 1
=54 seconds to return upstream, so that his total time
will be 6+54=60 seconds, which is precisely the time
taken by T.
This hypothetical phenomenon is called the " Fitzgerald
Contraction." Its value depends, of course, on the velocity
of the stream ; when the stream runs at 4 feet per second
and the speed of the rowing is 5 feet per second, the con
traction ratio has been shown to be x/( 1 —%)= x /(~)
=. The reader will see, from this way of writing it,
what its value would be in other cases.
In 1905, an alternative explanation was offered by
Einstein.
Einstein's Hypothesis.
Einstein lays down two general principles or axioms : * —
(i) It is impossible to detect uniform motion through
the ether. *J~>n
(ii) In all forms of wave motion, the velocity of propaga I
tion of the wave is independent of the velocity off
th© source. ^
Let us consider what these axioms mean.
(i) There is no difficulty in measuring the velocity of
one body relative to another : all our ideas of velocity are
essentially ideas of relative velocity, either velocities of
  . f\
THE VELOCITY OF LIGHT 31
other things relative to ourselves or our own velocity
relative to something else — e.g. a man who looks at the
road along which he is driving his car is probably estimating
his own velocity relative to the road. But it is meaning
less to inquire what our velocity is relative to the ether ;
no part of the ether can be distinguished from any other
part : it may be possible to identify matter in the ether,
but the ether itself defies identification. And if the ether
cannot be (so to speak) labelled anywhere, the statement
that a body is moving through it carries no information
with it, or in other words has no meaning attaching
to it.
(ii) The second axiom is perhaps more tangible. Imagine
an engine moving at a uniform rate along a straight railway
line on a perfectly calm day. If the enginedriver throws a
stone forwards, a man on the line will observe that the
velocity of the stone is equal to the velocity given it by
the thrower+the velocity of the engine. The faster the
engine is moving, the faster the stone will move, although
the thrower exerts the same effort as before. The velocity
of the stone relative to the air, therefore, depends on the
velocity of the source, namely, the man on the engine.
Suppose now the engine whistles, and is heard by a man
farther down the line. We know that the sound travels
in the form of a wave through the air at approximately
1100 feet per second. But the motion of the soundwave
is a different type of motion from that of the stone : its
velocity of propagation through the air does not depend
on the velocity of the engine at the moment it whistled,
i.e. it does not depend on the velocity of the source. The
speed of the train will affect the pitch of the sowidwave,
its musical note ; but the time the wave takes to reach
the man is not affected by the rate at which the engine
is moving. If, then, a particle in motion sends out a beam
of light, the rate of propagation of^the lightwave through
32 READABLE RELATIVITY
the ether has no connection with the velocity of the
particle which emitted the beam of light.
300,000 km.
200,000 km.
lOO.OOOkm. !
Fig. ii.
P and Q represent two places 300,000 km. apart and
rigidly connected together. I take up my position at P
and send a ray of light along PQ and measure the time it
takes to reach Q. If PQ is fixed in the ether (assuming
for the moment this phrase has a meaning), the time will
be 1 second. If, however, my observations give the time
as (say) only § second, I can calculate that the ray itself
only advanced §X 300,000 =200,000 km. through the ether,
and that therefore Q must have advanced 300,000—200,000
=100,000 km. to meet it in the same time, f second. Con
sequently the rigid bar PQ is moving at the rate of
ioo,ooo4f=i5o,ooo km. per second. But as I remain
at P, I deduce that my velocity through the ether is also
150,000 km. per second. This, however, contradicts
Axiom (i), which lays down that a discovery of this nature
is impossible. We are, therefore, forced to conclude that
the measurement of the time of flight over this distance
will always under all conditions be 1 second. Einstein's
two axioms taken together, therefore, involve the following
important result :
Any one who measures experimentally the velocity of light
in a vacuum will always obtain the same result (within, of
course, the limits of error imposed by the experiment). The
velocity of light in a vacuum, as determined by every individual,
is an absolute constant.
THE VELOCITY OF LIGHT 33
This conclusion may well cause a shock to any one who
considers carefully what it implies ; and the shock will not
be diminished by examining its bearing on the problem of
the boats.
It is, of course, important to notice the fundamental
distinction between lightwaves in the ether and sound
waves in the air. If an observer, when measuring the
velocity of sound, obtains an answer which does not agree
with the standard answer (about 1100 feet per second), he
can at once calculate his velocity through the air. Nor is
there any reason why he should not be able to do so ; and
he can compare his result with that obtained by other
methods. But with the ether it is otherwise ; the inability
of an observer to measure his velocity through the ether
involves the fact that his measure of the velocity of a light
wave must agree with the standard measurement.
The Application of Einstein's Hypothesis.
No one can be conscious of moving through the ether.
An onlooker O has no difficulty in measuring the speed
at which a man L is moving away from him ; it will be equal
and opposite to the velocity with which L calculates that
O is moving away from him : if each expresses his measure
of this velocity as a fraction of the velocity of light, the
results obtained by O and L will be numerically equal and
opposite in sign. Relative velocities, therefore, present
no difficulty. But O and L alike will each consider himself
at rest in the ether and will make his own measure
ments on that assumption. They, therefore, must be
regarded as looking at the world from different points
of view.
To explain the enigma of the boats, we must, therefore,
consider separately the standpoint of each of the actors
in the drama, the oarsmen T and L and an onlooker O,
whom we will regard as poised just above T and L at the
3
©
34 READABLE RELATIVITY
moment they start rowing. To make the analogy with
the M.M. experiment closer, imagine that the riverbanks
have disappeared, and that all we can see is an expanse
of mter devoid of all features or landmarks — that is what
the etheridea requires.
O says that this featureless ocean is
moving in the direction A>C at 4 feet
4 ft :j [s ec, per second ; in proof of this state
ment, he places a piece of cork on the
water and observes that it at once
(l) moves away from him in the direction
£ AC at 4 feet per second. T and L say
Fig. 12. that the water is motionless ; each,
sitting in his boat, places a piece of
cork on the water and it remains where they have placed
it ; O, of course, says that these pieces of cork are drifting
at the same rate as the boats. Further, T and L agree
that O is moving away from them in the direction C>A
at the rate of 4 feet per second ; they say that the piece
of cork which O has dropped remains stationary, and
that it is O who is moving away from it.
The statement that the velocity of light is an absolute
constant, or that each person who measures it obtains the
same answer as any one else, when applied to the boats,
means that T, L, and O will each obtain the same result
when they measure the speed at which each boat is
rowed through the water, because the boats are re
placing the lightrays in the M.M. experiment. We have
taken this common measure of the speed as 5 feet per
second.
Under these circumstances, our problem is to explain
a definite experimental observation, namely, the fact that
the boats (i.e. the lightrays) do return to A at the same
moment.
Regard AB and AC as rigid planks of wood floating on ,
THE VELOCITY OF LIGHT 35
the water. T and L believe that these planks are at rest,
just as they believe the water is at rest ; O believes that
the planks are drifting with the water, just as T and L
are doing.
T, L, and O each have a footrule ; T holds his SBbng
AB, and L holds his along AC. O compares his footrule
with T's by actual superposition, and they note that the
two rules agree. As long as T's rule is kept perpendicular
to the stream, it will remain identical with O's rule ; but
we shall see that when L, after comparing his rule with
T's, places it along AC in the line of the stream, will
consider it to contract although both L and T are un
conscious that it does so, and must remain unconscious of
this fact, because they can have no knowledge of the
existence of any stream carrying them along.
With the data of the problem, T and L satisfy themselves
by direct measurement that AB and AC are each 90 foot
rules long. T and L, neither of whom recognise the exist
ence of a stream, then calculate that their times to B,
2X90
C respectively and back to A will be m each case — z —
=36 seconds. And their clocks must bear this out when
the trips have been made, for otherwise they could infer
the influence of a stream and calculate its velocity.
O now times T's trip. By the argument on p. 26, he
sees that T makes a headway of 3 feet per second along
AB and back, and therefore takes 2 x 9 ° ==60 seconds for
the whole journey. Consequently O says that T's clock
only registers 36 seconds when it should register 60 seconds ;
therefore, according to O, T's clock loses.
Now L and T take precisely equal times for their trips.
Therefore, by O's clock, L also takes 60 seconds. But by
the argument on p. 26, O sees that L advances at 9 feet
per second from A to C, and returns from C to A at 1 foot
36 READABLE RELATIVITY
per second. Therefore if AC =90 feet, the total time
==^+^==10+90=100 seconds; but the total time
according to O is only 60 seconds.
/. according to O, the length of AC is only _ X90
=54 feet.
[As a check, note that ^4+54=6+54=6o seconds.]
y
It is true that L marked out AC by taking 90 of his foot
rule lengths ; therefore O is forced to conclude that L's
footrule is only ^=f foot long, and so O says that the
stream causes L's footrule, when placed along it, to contract
to I foot.
Further, as L also records the time of his trip as 36 seconds,
O says that L's clock loses at just the same rate as T's clock.
We may summarise these results as follows :
says that (i) clocks in the world of T and L lose
time ; they register an interval which is really
5 minutes long as only 3 minutes (60 : 36=5 : 3) ;
(ii) a footrule in the world of T and L measures
1 foot when placed along AB at right angles to
the stream, but only measures § foot when placed
along AC in the line of the stream.
T and L say that (i) their clocks keep normal time ;
(ii) their footrules remain 1 foot long, in what
ever position they are placed.
Who is Right ?
It seems absurd to suggest that all of them are right.
Let us, however, inquire what L thinks about O. Suppose
that O and his brother O' mark out two courses, AB and
AD, each of length 90 feet, along CA produced and AB, in
the air just above the ocean.
THE VELOCITY OF LIGHT 37
Then L says there is a current 'in the air of 4 feet per
second which carries O and O' in the direction C»A ;
4 ft. /sec.
^<
©
Q
A
Fig. 13.
©
O and O', of course, say that the air is at rest and that
L is drifting with the water in the direction A>C at 4 feet
per second.
Suppose now that O and O' fly through the air at 5 feet
per second (i.e. relatively to the air). O' flies to B and
back again to A, O starts at the same time as O' and flies
to D and back again to A. They both arrive back at A at
the same moment. This is the experimental fact estab
lished in the M.M. experiment, which needs explanation.
It is clear that L's views (or T's views) about O and O'
are precisely the same as those which O formed about T and
L. The arithmetical calculations are identical and need not
be repeated. The results may be expressed as follows :
L and T say that (i) clocks in the world of O and O'
lose time ; they register an interval which is
really 5 minutes long as only 3 minutes ;
(ii) a footrule in the world of O and O' measures
1 foot when placed along AB at right angles to
the current, but only measures f foot when placed
along AD in the line of the current.
and 0' say that (i) their clocks keep normal time ;
(ii) their footrules remain 1 foot long, in what
ever position they are placed.
38 READABLE RELATIVITY
It is clear, therefore, that any argument that can be
used to support the views of or 0' can be applied with
equal force to support the views of T and L. We must,
therefore, regard /both views as equally true. We are
therefore forced to conclude that each world, the world
of O, O', and the world of T, L, has its own standard of
timemeasurement and its own standard of lengthmeasure
ment. If one world is moving relatively to another world,
their standards of time and space automatically become
different.
Suppose two people come together and compare their
clocks to make sure they run at the same rate, and compare
their footrules to make sure they agree, and suppose that
afterwards they separate at a uniform rate, one from the
other, along a line AC. Now imagine two explosions to
take place at different times at different places somewhere
on AC. Each observer, making proper allowance for the
time sound takes to travel, can measure the timeinterval
between the two events and the distanceinterval of the
spots at which they occurred. But their measurements
will not agree, either as regards timeinterval or as regards
distanceinterval, for they have different standards of time
and different standards of length.
There is indeed one measurement about which they will
agree, namely, the velocity of a ray of light : each of them,
using his own clock and his own rule, will find experimentally
that a lightwave travels at 300,000 km. per second.
Note. — The statement (see p. 28) that further repeti
tions of the M.M. experiment have confirmed the conclu
sion that no etherstream can be detected requires some
qualification. Just recently, Professor Dayton Miller be
lieves that he has been able to measure a drift which
varies from zero at sealevel up to 10 km. per second at
the summit of Mount Wilson. The details of his experi
THE VELOCITY OF LIGHT 39
ments are not yet available in this country, but Professor
Eddington has argued {Nature, June 6, 1925) that a
differential drift of this kind is inconsistent with astrono
mical observations. It, therefore, appears probable that
Professor Miller's experiments may admit of an interpreta
tion different from that which he is reported to give.
EXERCISE III
1. A man's footrule is really only 10 inches long ; what is the
true length of a fence which the man measures as 12
yards ? What will the man say is the length of a fence
whose true length is 20 yards ?
2. A footrule contracts to £ of its proper length. What is
the true length of a line which according to this footrule
is y feet ? If the footrule is used to measure the length
of a line whose true length is z feet, what result is obtained?
3. O says that two events occurred at an interval of 12 seconds
at places 18 feet apart. What measurements are given,
by L, if his clock only registers 45 minutes for each hour
of O's clock, and if his footrule only measures 8 inches
according to O's rule ?
4. A stream flows at 3 feet per second, and a man can row at
5 feet per second through the water. The width of the
stream is 40 feet. Find the times taken to row (i) straight
across the stream and back, (ii) 40 feet downstream and
back.
5. With the data of No. 4, find how far the man can row down
stream and back if he takes the same time as he would to
go straight across and back.
6. It is found that a bullet from a rifle travels 1 100 feet in the
first second of motion. The bullet is fired along a rail
way line from a train at a moment when a man is 1 100 feet
away in the line of fire. There is no wind. Does the
bullet or the noise of the explosion reach the man first if
the train (i) is moving towards the man, (ii) is at rest,
(iii) is moving away from the man ?
7. A stream flows at u feet per second, and a man can row at
c feet per second through the water. The width of the
stream is x feet, and the man can row straight across
40 READABLE RELATIVITY
and back in the same time that he can row x x feet down
stream and back. Prove that —
<T> 2 * _ *! , *i
w *Jc 2 —u a c+u^c—W
(ii)* 1= *</(l_" 2 )
CHAPTER IV
CLOCKS
" Alice looked round in great surprise. ' Why, I do believe
we've been under this tree all the time ! Everything's just
as it was ! '
" ' Of course it is,' said the Queen ; ' what would you have
it ?'
" ' Well, in our country,' said Alice, ' you'd generally get to
somewhere else if you ran very fast for a long time, as we've
been doing.'
" ' A slow sort of country ! ' said the Queen. 'Now here,
you see, it takes all the running you can do to keep in the same
place. If you want to get somewhere else, you must run at
least twice as fast as that ! ' " — Through the LookingGlass.
Observations at Different Places.
If several observers, who are recording the times of
occurrence of a series of events, wish to exchange their
results, it is necessary for them to compare their clocks.
Preferably the clocks should be synchronised, but it would
be sufficient to note the difference between each clock and
some standard clock. The standard British clock registers
what we call " Greenwich time."
Synchronising is a simple matter if the observers and their
clocks are all at one place, but if the observation stations are
far apart direct comparison is impossible, and we are forced
to rely on indirect methods which may not be proof against
criticism. To transport a clock from one station to another
is not a reliable method, because the journey itself may
set up an error in the running of the clock. The best
method is to send signals from a standard station to all
42 READABLE RELATIVITY
other stations, and use these signals to synchronise the
various clocks or record their errors ; this in fact is done
each day by the wireless signals sent out at noon from
Greenwich. Wireless signals travel with the velocity of
light, and therefore, for such comparatively small distances
as we are concerned with on the Earth, the time of transit
of the signal is usually negligible. But for large distances,
such as the distance of the Sun from the Earth, the time
taken by the signal is material, and allowance must be
made for it in setting the clock. We shall see, however,
that the process involves another difficulty which we are
powerless to remove. This is best illustrated by a numerical
example. In order to avoid big numbers and to bring the
arithmetic of this chapter into line with that of the last,
we shall introduce (temporarily) a new unit of length :
60,000 km.=i leg.
The velocity of light is therefore 5 legs per second.
Synchronising Two Clocks.
Suppose that two observers A and C, relatively at rest
to each other, are at a distance of 75 legs apart, as measured
by their own rules ; this distance is about twelve times as
much as the distance of the Moon from the Earth. We
shall examine the process by which A and C attempt to
synchronise their clocks.
5< 7
75 legs
>5
Fig. 14.
Since light travels at 5 legs per second, A and C calculate
that a ray of light sent by either to the other will take
25 =15 seconds to travel across the space separating them.
CLOCKS 43
It is agreed that, at the instant when A's clock records zero
hour, A shall send a lightsignal to C, and that C, immedi
ately he receives it, shall reflect it back to A.
C therefore sets his clock at 15 seconds past zero, but does
not start it until the signal from A arrives. Immediately
C receives the signal, he starts his clock and believes that
it now agrees with A's clock. This opinion is shared by A,
who, when he sees his clock indicating 15 seconds past
zero, says to himself, "At this moment C is receiving my
signal." From A's point of view, the fact appears to be
established beyond doubt when the return signal from C
reaches A at the instant his (A's) clock registers 30 seconds
past zero. We know that A's receipt of the return reflected
signal must occur at this instant, because otherwise A would
be able to calculate his velocity through the ether (compare
p. 32), and this, as we have seen, is impossible. In the
same way, C, when he sees his clock indicating 30 seconds
past zero, says to himself, " At this moment A is receiving
the return signal," and this opinion is confirmed by the
fact that, if A then reflects the signal back to C, it will reach
C when C's clock indicates 45 seconds past zero, for the
same reason as before.
Now there can be no ambiguity as regards the time indi
cated by C's clock of an event happening to C, nor as
regards the time indicated by A's clock of an event hap
pening to A. But we shall see that there is unfortunately a
great deal of uncertainty as to the time indicated by A's
clock of an event happening to C, or vice versa. If the
clocks of A and C are genuinely synchronised, this uncer
tainty would not exist. But if there are grounds for
suspecting that A and C are mistaken in their belief that
they have succeeded in synchronising their clocks, there is
no direct method of either of them ascertaining the time by
his own clock of an event which is happening to the other.
Although, when A sees that his clock reads 15 seconds past
44 READABLE RELATIVITY
zero, he says that at this moment his signal is arriving at C,
yet he has no direct method of making sure that this state
ment is true. And, by enlisting the evidence of an eye
witness, we shall show that there are different, but equally
trustworthy, opinions of the time recorded by A's clock of
the arrival of the signal at C.
An Onlooker's Opinion.
We now introduce an onlooker O, who considers that
the world of A and C is moving away from him in the
direction A>C at 4 legs per second.
Each individual acts on the supposition that he himself is
at rest. In the following inquiry into O's opinions, we
must therefore regard O as at rest and A, C as moving
away from O. But if we had to inquire into the views of
A or C, we should have to regard them as at rest and O
as moving away from each of them in the opposite
direction.
Suppose that A is passing O at the moment when A
sends out his first light signal, and that O also sets his
clock at zero at this instant. We can connect O with the
world of A, C most easily by imagining that A combines
his timesignal with a performance of the M.M. experiment.
A marks out a track AB at right
angles to AC and makes it 75 legs
long by his rule, and places a mirror
4 legs pe r sec at B in the usual way. At the
*"*" same time as he sends his light
signal to C he sends another to B,
and, as we know, both rays, re
0,A 75 r flected back, return to A at the
_, same moment.
Fig. 15. ^ T
Now O and A agree that the
length of AB is 75 legs, because for lengths across the
stream their rules are identical. Also O, A, and C all
75
CLOCKS
45
0,A
agree that light travels at 5 legs per second through the
ether.
Fig. 16 represents O's idea of the path pursued by the
light signal, which is directed to the mirror at B.
By the time the light
signal impinges on the J^. Jji
mirror at B, the arm
AB has moved to the
position AiB x , so that
the signal starts from
O, A and impinges on the
mirror at B v and there
fore travels along OBj ;
by the time it returns to
A the arm AB has moved to the position A 2 B 2 , so that the
return path is BjA 2 .
The arm AB is advancing at 4 legs per second and the
light signal travels along OB x at 5 legs per second. Suppose
the outward journey takes t seconds. Then OB^s* legs,
BB X =4* legs, OB =75 legs,
/. by Pythagoras, (5*) 2 =(4*) 2 +75 2 
.. 25P— 162 2 =75 2 or 9* 2 =75 2 or 32=75.
.*. t= L ^=2S seconds.
3 D
.'. the total time out and back according to O's clock is
2x25 =50 seconds.
But by the M.M. experiment the ray returns to A from C
at the same moment as the ray from B.
.*. by O's clock, the ray returns to A from C at 50 seconds
past zero.
But A's clock registers 30 seconds past zero when the ray
returns to A from C.
.*. the arrival of the ray back at A is said by O to occur
at 50 seconds past zero by O's clock, and to occur at
30 seconds past zero by A's clock.
46 READABLE RELATIVITY
Therefore, although O's clock and A's clock agreed at
zero hour, they do not agree afterwards: we may there
fore say that the synchronisation between O and A has
disappeared.
Let us next ascertain O's opinion as to the time when the
first signal reached C.
O says that the ray from A to C is advancing at 5 legs
per second towards a target C which is retreating at 4 legs
per second : the ray therefore gains on the target C at
5—4=1 leg per second. But on the return journey the
ray advances at 5 legs per second towards a target A
which advances to meet it at 4 legs per second : the ray
therefore gains on the target A at the rate of 5+4=9
legs per second. The distance which the ray has to gain
on its target is the same on each journey (A and C
believe this distance is 75 legs; O does not agree
with them : but we need not stop to ascertain O's
estimate of the distance), therefore the outward journey
A>C takes 9 times as long as the journey back from G to
A, so that x^ths of the total time is spent on the outward
journey, and T l ^th of the total time on the return journey.
Now the total time, out and back, by O's clock is 50
seconds.
.*. O says the outward journey, A>C, takes ^th of 50
=45 seconds, and the journey back from C to A takes
J^th of 50 =5 seconds.
.*. says that the ray arrives at C at 45 seconds past
zero by O's clock.
Also since A's clock records the total time, out and back,
as 30 seconds, we see in the same way that says that the
ray arrives at C at ^th of 30=27 seconds past zero by
A's clock.
Further, when the ray arrives at C, we know that C's
clock registers 15 seconds past zero and is set going at this
instant.
CLOCKS
47
The occurrence of the event consisting of the arrival of
the ray at C is therefore registered by as follows :
O's Clock.
A's Clock.
C's Clock.
45 seconds past zero.
27 seconds past zero.
15 seconds past zero.
This is O's opinion of the operation. A of course does not
agree with O ; when A's clock registers 27 seconds past
zero, A says that it is long past the time of C's receipt of
the signal.
O, however, says that C's clock has been set 27—15
=12 secondspaces behind A's clock.
We can easily continue this process of calculating the
times registered by O of further events. Consider the
arrival of the ray back at A from C.
C dispatches the ray to A and receives it back again
2 X 7^
reflected from A after a total interval of — ^=30 seconds,
by C's clock.
Now O says that the time from C to A is only j^th of the
total time C>A and A>C.
Therefore O says that the ray takes ^th of 30 =3 seconds,
by C's clock, to travel from C to A. But the time on
C's clock when the ray left C was 15 seconds past zero ;
therefore the time on C's clock when the ray arrives at A
is 15+3=18 seconds past zero, according to 0.
The occurrence of the event consisting of the arrival of
the ray back at A is therefore registered by as follows :
O's Clock.
A's Clock.
C's Clock.
50 seconds past zero.
30 seconds past zero.
18 seconds past zero.
48 READABLE RELATIVITY
•It is worth comparing these two events as recorded by 0.
O's Clock.
A's Clock.
C's Clock.
Event I. (arrival
atC)
Event II. (return
to A)
45
5o
27
3°
15 seconds past
zero.
18 seconds past
zero.
Time interval be
tween the events
5
3
3 seconds.
therefore says that A's clock and C's clock run at the
same rate (each registers the interval between the two
events as 3 seconds), but both their clocks lose time (each
records an interval as 3 seconds which is really 5 seconds
long) and C's clock has been set 12 seconds behind A's clock.
What other Onlookers Think.
Now these calculations which O has made have depended
on the fact that the world of A, C is moving away from O at
4 legs per second. Suppose that there is another onlooker
P, who notes that the world of A, C is moving away from him
in the direction A»C at (say) 3 legs per second. Then the
same argument which has been used to obtain O's records
may be used to obtain P's records of the various events,
but the arithmetic will be different, and P's opinion about
the behaviour of the clocks of A and C will not agree
numerically with O's opinion. P will say that A and C
have failed to synchronise, but will form a different esti
mate of the amount C's clock is behind A's clock, and will
assess at a different figure the rate at which both A's clock
and C's clock lose. It is left to the reader to make the neces
sary calculations, see Exercise IV., No. 2. Each onlooker,
therefore, has his own standard of time ; and his judgment
of the timeinterval separating two events will differ from
CLOCKS 49
that formed by another observer moving relatively to him.
This agrees with what has been said in the previous chapter.
But the example we have just taken shows also that it is
impossible to synchronise two clocks which are situated
at different places. For, although the inhabitants of the
world in which the clocks are at rest believe that they have
secured synchronisation, the observers in other worlds not
only deny that they have done so, but disagree amongst
themselves as to the amount of the difference between the
clocks. No setting of the clocks can therefore ever secure
general approval, or indeed approval by the inhabitants of
more than one world.
Simultaneous Events.
If after A and C believe they have synchronised their
clocks, an event takes place at A and another event takes
place at C, and if each records the time of the event which
has happened to himself, and if these two records are the
same, then A and C will say that the two events happened
simultaneously. But with the data of our example, we
see that O will say that the event at A took place before
the event at C, for according to O when A's clock reads
27 seconds past zero C's clock reads 15 seconds past zero.
Therefore if A and C both say that the times of the events
are 27 seconds past zero, O says that, when the event occurs
at A, C's clock has only got as far as 15 seconds past zero
and therefore the event at C, timed as 27 seconds past
zero at C, has not yet occurred. In fact the timeinterval
between these two events is 27— 15 =12 seconds as measured
by the clockrate of A or C, which is equivalent to 20 seconds
as measured by the clockrate of O, for 5 of O's seconds
are the same as 3 of A's seconds or C's seconds. O will
therefore say that the event at A took place 20 seconds
(by O's clock) before the event at C.
A and C therefore call two events simultaneous which
4
50 READABLE RELATIVITY
O considers occur at a definite timeinterval, and other
onlookers will agree with O in saying that the events are
not simultaneous, but will disagree with O as to the length
of the timeinterval between them. It is therefore im
possible to attach any meaning to a general statement
that two events at different places occurred at the same
time. If the timestandard of one world makes them
simultaneous, the timestandard of other worlds require
a timeinterval between the events. Since there is no
reason to prefer the opinion of one onlooker to that of any
other, we cannot say that any one opinion is more correct
than any other. The bare statement that two events at
different places were simultaneous is, therefore, devoid
of meaning, unless we also specify the world in which this
timemeasure has been made.
Union of Space and Time.
Time by itself ceases to be an absolute idea ; it is a pro
perty of the world in which it is measured, and each world
has its own standard.
Each individual has, of course, his own timerule and
his own distancerule which he thinks of as absolute,
because he thinks of his own world as at rest. But in a
sense this is a delusion, because a transference to another
world will modify each of them ; a change in timemeasure
is bound up with a change in distancemeasure. As we
have already seen, the onlookers do not agree with A, C,
or each other as to the distance between the two places
where the events occurred, any more than they agree with
the timeinterval between the events. To quote the
celebrated phrase of Minkowski : " From now onwards
space and time sink to the position of mere shadows, and
only a sort of union of both can claim an independent or
absolute existence " — i.e. an existence to which all onlookers
will give equal recognition and apply equal standards of
CLOCKS 51
measurement. We shall see later what form this union
takes.
EXERCISE IV
1. A and C measure their distance apart as 50 legs ; an on
looker P notes that the world of A, C is moving away from
him in the direction A>C at 3 legs per second. A passes
P at zero hour by A's clock and P's clock, at which
moment A sends a light signal to C in order to synchron
ise with C ; this signal is reflected back to A. What is
P's estimate of the times recorded on the three clocks of
(i) the arrival of the signal at C, and (ii) the return of the
signal to A ?
2. Repeat No. 1, if A and C are 75 legs apart.
3. With the data of No. 1, D is a person in the world of A and
C at a distance of 100 legs from A and on the other side
of C from A. If A and D synchronise, find the difference
between their clocks according to P in terms of second
spaces (i) on A's clock, (ii) on P's clock.
4. Repeat No. 1, assuming that the world of A, C is moving
away from P in the direction C>A at 3 legs per second,
P and A being as before at the same place at zero hour.
5. With the data of No. 1, if an Event I. occurs at A and an
Event II. occurs at C, and if A and C describe these
events as simultaneous as recorded by their own clocks,
which event will P consider to have occurred first ?
Repeat this problem with the data of No. 4.
6. Two events, I., II., occur simultaneously at different places
in the world of A and C. An onlooker O says that I.
occurred before II. Would it ever be possible for some
other onlooker to say that II. occurred before I. ?
7. With the data of No. 1, find P's estimate of the distance of
A from C.
CHAPTER V
ALGEBRAIC RELATIONS BETWEEN TWO
WORLDS
"The progress of Science consists in observing inter
connections and in showing with a patient ingenuity that the
events of this evershifting world are but examples of a few
general relations, called laws. To see what is general in what
is particular, and what is permanent in what is transitory,
is the aim of scientific thought." — A. N. Whitehead, An
• Introduction to Mathematics.
Generalisations.
We have shown in the previous chapters, by means of
numerical examples, that any eyewitness will consider
that standards of measurement of distance and time vary
from one world to another. The real nature of these
variations cannot be appreciated unless we pass on from
numerical illustrations to general formulae. We therefore
shall now proceed to express in algebraic form the relations
betweei^ two worlds which are moving with uniform
velocity relatively one to another. These formulae may
then be utilised for solving special numerical cases.
It will simplify the work if we introduce a new unit of
length :
300,000 km. (i.e. 5 legs) =1 lux.
The velocity of light is therefore 1 lux per second.
Statement of the Problem.
It may assist the reader if we state in great detail the
problem proposed for solution in this chapter.
53
54 READABLE RELATIVITY
The world of A and C is moving away from O in the
direction A^C at a uniform velocity of u luxes per second ;
at the instant when A passes O, both A and O set their
clocks at zero hour. A and C are at rest relatively to each
other, and they measure their distance apart as x x luxes.
A and C believe they have synchronised their clocks.
An event (Event I.) occurs at A at zero hour by A's
clock ; another event (Event II.) occurs at C at t x seconds
past zero by C's clock. Therefore in the world of A and C
the distanceinterval between the two events is x x luxes,
and the timeinterval between the two events is t x seconds.
There is complete agreement between A and C as to both
of these interval measurements. Each regards both him
self and the other as at rest in the ether. Their distance
measures agree because they can use the same rule to measure
out AC ; their timemeasures agree, because otherwise
they could deduce the velocity of their common world
through the ether.
Next consider O's point of view. He says that Event I.
occurs at O at zero hour, and that Event II. occurs at C
at (say) t seconds past zero by his own (O's) clock. O regards
himself as at rest and A, C as moving away from him.
O therefore takes the distanceinterval between the two
events as the distance of C from him at the moment when
Event II. takes place. Suppose that O's measure of this
distance is x luxes. Then says that the distanceinterval
between the two events is x luxes and the timeinterval
between the two events is t seconds.
In short, the interval between the two events is registered
by A or C as x x luxes, t x seconds, and by O as x luxes,
/ seconds.
What are the formulae which connect x, t with x x , t x ?
Before tackling this general problem, we shall ascertain
O's opinion about the measuringrule used by A or C, the
running of their clocks, and their attempts to synchronise.
ALGEBRAIC RELATIONS
55
MeasuringRules.
A marks out a length AC of x x luxes along the line of
motion of A's world relatively to O. What is the length
of AC, according to O ?
B B, B 2
v luxes per sec.
■»
QA C
Fig. 17.
O.A
Suppose that O is watching A performing the M.M. experi
ment. A and C agree that the lengths of AC and AB are
each x x luxes, O agrees that the length of AB is x % luxes,
but says that the length of AC is different, say z luxes.
says that the arm AB moves away from him at u luxes
per second, so that the ray sent towards the mirror at B
impinges on it when AB has moved into the position
AjBi ; the path of the ray is therefore AB r Similarly,
the ray returns to A when AB has moved into the
position A 2 B 2 , so that the return path is B^.
Suppose the time from A to B x or from B x to A 2 ,
is k seconds by O's clock. O makes the following
calculations :
ABi=^ luxes (light travels along AB X at 1 lux per second).
BB ± =ku luxes (AB advances at u luxes per second).
AB =x x luxes (O agrees with A's measurements across the
stream).
.*. by Pythagoras, k 2 =k?u 2 \x 1 2
:. W—Wu*=x? or k 2 (iu 2 )=x 1 z
k*=
I— M a
56 READABLE RELATIVITY
The total time by O's clock from A to B and back is
2k seconds.
.*. the total time by O's clock from A to C and back is
2k seconds.
But O can also reason as follows :
From A to C, the ray travels at i lux per second towards
a target C, z luxes away, which is retreating at u luxes per
second. Therefore the ray gains on the target at (i— u)
luxes per second.
.*. the time from A to C by O's clock is seconds.
i — u
Similarly from C to A the ray travels at I lux per second
towards a target A, z luxes away, which is advancing at
u luxes per second. Therefore the ray gains on the target
at (i+w) luxes per second.
.*. the time from C to A by O's clock is — ; — seconds
ijw
.'. the total time from A to C and back is — — + — ^— seconds
I — U IjM
z(i+u)+z (i— U) _Z+ZU+Z— ZU 2Z ,
 (i w )(i +w ) f=^2 =£Z^r* seconds.
But the total time by O's clock is 2k seconds.
2Z
.*. i _ m2 =2& or z=k(i— u 7 )
;. 2 2 =& 2 (iw 2 ) 2 , but k 2 =^~ 2
••• * a =^2(l« 2 ) 2 =*! 2 (l« 2 )
.". z=x 1 V(i— u 2 ).
Therefore says that a length in the direction of motion
which A and C measure as x x luxes is really x x V{l— u 2 )
luxes :
Or, in proportion, what A and C measure as I lux is in O's
opinion really V(i— u 2 ) luxes.
ALGEBRAIC RELATIONS 57
Now V(i— u 2 } must be less than i ; consequently says
that the measuringrule used by A and C, when placed
along the line of motion, contracts ; and the contraction
ratio is V{i— m 2 ).
Comparison by O of A's Clock and C's Clock with O's
Clock.
Our numerical examples have shown that every one
will agree that A's clock runs at the same rate as C's clock.
The reason for this may be stated as follows :
An essential feature in every argument is that each
individual regards himself as at rest in the ether and that
all the observations he makes must bear this out. He
cannot make any measurement which will reveal his
velocity through the ether. A and C agree that their
distance apart is x x luxes : they therefore argue that a
time signal sent from either to the other and reflected
back will return after 2X X seconds, and their clocks must
bear this out. But the experiment in which A sends a
signal to C and receives it back again is identical with the
experiment in which C sends a signal to A and receives it
back again. Both A's clock and C's clock record the time
of this experiment as performed by each of them as
2% seconds. Therefore A's clock and C's clock must run
at the same rate. We have seen from numerical examples
that O admits this, but says that both clocks lose and that
they have not been synchronised. Let us now calculate
O's estimate of the timedifference between A's clock and
C's clock.
In order to synchronise the clocks, A proposes at zero
hour by his clock to send a lightray to C. As they agree
that AC is x x luxes, they calculate that the signal will take
x x seconds to reach C. Consequently C sets his clock at
x x seconds past zero and starts it at the instant the lightray
arrives.
58 READABLE RELATIVITY
Now we have just seen that by O's clock the time from
A to C is  — m seconds, and the total time out and back is
i — u
2.Z
.^—^2 seconds. O therefore says that the fraction of the
total time out and back occupied by the journey out is
Z _^_ 2Z _ Z (l\u)(l—u) _ L\U
I — U ' I—U 2 ~~I — U 2.Z 2 "
Now A's clock registers 2x x seconds for the total time out
and back. Therefore O says that, when the ray arrives
i I ii
at C, A's clock registers — — X2x 1 =x 1 (i\u) seconds past
zero. But at this instant C's clock starts off at x ± seconds
past zero.
.*. A's clock is ahead of C's clock by x x {~l\u)—x x seconds
=x 1 \x 1 u— x 1 =x 1 u seconds.
Therefore when A and C think they have synchronised
their clocks, O says that A's clock is x x u secondspaces
ahead of C's clock.
The difference between the clocks depends on the value
of x v the length of AC. Therefore the farther C is away
from A in the direction of motion of the world of A, C
from O, the more A's clock is ahead of C's clock, according
to O. Suppose, for example, the world of A, C is moving
due east away from O. Then A's clock is ahead of any
clock east of A and is behind any clock west of A. Both
these results are expressed in the statement given above,
because, if C is west of A, x x is negative, and a clock which
is a negative number of seconds ahead of another clock is,
of course, a positive number of seconds behind it.
We must therefore regard each place on the line of
motion of AC as having its own clock : the inhabitants
of the world of A, C think all these clocks are synchronised,
but O says each registers a local time whose difference
ALGEBRAIC RELATIONS 59
from that of A is given by the formula above. We may
express the facts by a diagram showing the local time,
according to O, of the instant when A is passing O,
C x 2 luxes
u luxes per sec.
— »
x, luxes
x 2 u sec. after zero A x,u sec. before zero
zero hour
x
O
Fig. 19.
which is taken as zero hour both by A and O. The
distances indicated in the diagram represent A's or C's
measurements.
ClockRates.
A and C record the timeinterval between two events
as 1 second. What is O's estimate of this timeinterval
by his own clock ?
With our previous notation, we know that O says that
the time from A to C and back is 2k seconds by O's clock,
where & 2 == x 2 or k= ,. x 42V
1— u 2 v(i— u z )
But A says that the time from A to C and back is
2% seconds by A's clock and O must agree with him.
.*. O says that 2x x seconds on A's clock measures the same
timeinterval as 2k seconds = //t *<,< seconds on O's
V (I— ur)
clock.
.*. O says that (in proportion) 1 second on A's clock measures
the same timeinterval as —tj o\ seconds on O's
v(i— u*)
clock.
It is important to remember that this is a statement of
O's view about the behaviour of A's clock.
60 READABLE RELATIVITY
Since V(i— u 2 ) is less than i, / / 2 ^ is greater than i ,
and therefore says that A's clock loses. But, of course,
A equally says that O's clock loses. Our results always
depend on the point of view from which the progress of
events is being observed.
Time and Distance Intervals between Two Events.
The data which determine the two events have been
stated in great detail on p. 54. The diagram represents
O's view of the events.
(I.) Position at zero hour —
0,A
» it luxes per sec.
(II.) Position at t sec. past zero by O's clock —
1 .
O AC
Fig. 20.
Event I. occurs at zero hour at A at the instant when A is
passing O. Event II. occurs at C at t seconds past zero by
O's clock. A and C say that the distanceinterval between
the two events is x x luxes, i.e. their measure of AC is x x luxes.
O says that Event I. occurred at O, and that Event II.
occurred at the position of C at t seconds past zero by O's
clock. O therefore says that the distanceinterval between
the two events is x luxes, which is his measure of the length
of OC in (II.). O also says that the measure of OA in (II.) is
ut luxes, because A is moving away from him at u luxes per
second.
.*. O says that by his rule AC=x—ut luxes.
Now A measures AC as x x luxes and O says that a measure
ment of 1 lux by A is really v/(i— w 2 ) luxes (see
ALGEBRAIC RELATIONS 61
p, 56). Therefore O says that AC is really x x V(i— u 2 )
luxes.
.'• x 1 V(i—u 2 )=x—ut
x—ut
This relation is very important.
Again, suppose that the time of Event II. at C is recorded
by C's clock as t x seconds past zero. Then A and C must
agree that the timeinterval between the events is t x seconds.
Now O says that A's clock is x x u seconds ahead of C's
clock (see p. 58). Therefore O says that, when Event II.
occurs, the time on A's clock is t x +x x u seconds past zero.
But we know that 1 second on A's clock measures the same
timeinterval as ^,*_ u *\ seconds on O's clock.
.*. when Event II. occurs, the time on O's clock is
J^t^L seconds past zero ; but the time on O's
V(i— u 2 ) r
clock is t seconds past zero,
. , t x +x x u
" *V(lM 2 )
x — ut
;. t x +x x u=tV(iu 2 ) ; now %= ^ (l _^ 2)
. t W/ T „2\ UJXUt) t(lU 2 )u{xUt)
" ^l=^(Iw 2 ) v(I _ w2) V(I1*) —
_t— uH— ux\u 2 t
~ V(iu 2 )
_ t—ux
''• tx ~V{l.U 2 )
This relation is also of great importance. It may be a
help to the reader if we restate what has been established.
Two events occur at distanceinterval x x luxes and time
interval t x seconds according to A, C, and at distanceinterval
62 READABLE RELATIVITY
x luxes and timeinterval t seconds according to 0. The
world of A, C is moving away from the world of O at u luxes
per second, and distances are measured as positive in the
direction of motion of A from O. Then A's records are
connected with O's records by the formulae :
_ x—ut _ t—ux
If then we know the distance interval and the timeinterval
between two events as recorded in one world, we can cal
culate the distance and timeintervals between these two
events as recorded in any other world moving with uniform
velocity relative to the former, along the line joining the
two events.
A's Opinion of O's Records.
It has been pointed out frequently in previous chapters
that there is no observer whose records are entitled to more
respect than those of any other observer. It is therefore
essential to show that the formulae just obtained are con
sistent with this view. Using the same notation and axes
as before, A says that O is moving away from him at (— u)
luxes per second. Now A says that the distance and time
intervals between the events are x x luxes and t x seconds.
.*. the formulae just obtained show that
O's distanceinterval should= J 7 7 ^ ^=77 V\
v(i— w 2 ) V(i— u 2 )
and O's timeinterval should =^7/ 4r= A — ^
V(i— w 2 ) a/(i— m 2 )
.*. the formulae just obtained should be equivalent to
_ x x \ut x t _ t x +ux x
X V{IU*) anCl r ~A/(lM 2 )
Unless they' are, there is not that reciprocal relation
ALGEBRAIC RELATIONS 63
between O and A which the Theory of Relativity requires.
We may state the problem as follows :
x—ut , t—ux
Given thatx 1 = ^^ 1 __ uV j and h=^( 1 ^tf)
Phw that x^ v x ^._ uK) and *= V(l _^.
a;— ut u(t—ux)
(i) We have *i+«<i= v < (i __ < ^ +V(i < ^
#— ut+ut— u 2 x _ x(i—ti?)
 V(iu 2 ) ~V(iu 2 )
=xV{iu*)
•'• *</(lM 2 )'
(ii) We have <i+«* 1 = V( I _ < ,2) + ^(i^ 2 )
f— «3g+«x— «* a ^ __ l(i — u 2 )
= V(iw 2 ) V(i  w 2 )
=tV{iu 2 )
t x \ux x
V(lM 2 )*
2 =
We therefore see that the relations which express A's
opinion of O's world are consistent with, and can be deduced
from, the relations which express O's opinion of A's world.
The Velocity of Light.
The formulae which connect the two worlds introduce
the expression V(i— w 2 ), which is imaginary if «>i, i.e. if
the velocity of one world relatively to the other is greater
than the velocity of light. We therefore say that we can
have no experience of a body moving with a velocity greater
than that of light. And in all our results u must stand for
64 READABLE RELATIVITY
a fraction between +i and — i. It is customary to repre
sent V(iu 2 ) by i or to put P= v ,*__ uS , so that #>i.
In this case, the standard formulae may be written :
% 1 =P(x—ut) ; t 1 =ft(t—ux)
or
x=&{x x +ut^ ; t=/3{t 1 +ux 1 )
where P=tt «>*•
V(i—u 2 )
And the results on pp. 5960 may be stated as follows :
(i) O says that the length of a line in the direction of
motion which A measures as 1 lux is ^ luxes.
(ii) O says that a timeinterval which A's clock records
as 1 second is /3 seconds.
EXERCISE V
1. The world of A is moving at § lux per second due east from
O. What is O's opinion about (i) the length of A's
footrule, (ii) the rate of running of A's clock ? What is
A's opinion about O's footrule and O's clock ?
2. A and C, who are relatively at rest at a distance apart of
5 luxes, have synchronised their clocks ; the world of
A, C is moving away from O in the direction A>C at
ft lux per second. A passes O at zero hour by O's
clock and A's clock. What does O say is the difference
between A's clock and C's clock ? D is a place in the
world of A, C, such that DA = 10 luxes, DC = 15 luxes.
What does O say is the difference between D's clock and
A's clock ? What does O say is the time recorded by the
clocks of A, C, D when O's clock records 25 seconds past
zero ?
3. With the data of No. 1, A records two events as happening
at an interval of 5 seconds and at a distance apart of
3 luxes, the second event being due east of the first
event. What are the time and distanceintervals of the
events as recorded by O ?
ALGEBRAIC RELATIONS 65
4. With the data of No. 3, solve the question if the second
event is due west of the first event.
X— Ut . Xx + Utx
5. Given that *i= ,/ — ; and* = . ,
t—ux , t 1 +ux 1
Prove that U= . — =^andf = <7 v
1 VI M a VI— M a
6. If Event I. is the dispatch of a lightsignal by A and
Event II. is the receipt of the lightsignal by C, show
that with the usual notation (i) x 1 =t 1 , (ii) x=t. What
does this mean in terms of O's opinion ?
7. Using the equations on p. 63, prove that x*—t* is always
equal to x x 2  tj 2 .
CHAPTER VI
THE SEPARATION OF EVENTS
" ' That's the effect of living backwards,' the Queen said
kindly ; ' it always makes one a little giddy at first, but there's
one great advantage in it, that one's memory works both
ways.'
"'I'm sure mine only works one way,' Alice remarked.
' I can't remember things before they happen.'
" ' It's a poor sort of memory that only works backwards,'
replied the Queen." — Through the LookingGlass.
The position of an event in history is fixed if we know
(i) when, and (ii) where it took place. These necessary data
must be expressed with reference, at any rate implicitly,
to some standard event. The " when " is usually referred
to the Christian Era, t years a.d. ; the " where," if
on the Earth's surface, may be described in terms of
longitude and latitude, the standard being the Greenwich
meridian and the Equator. Astronomers often state
the " where " of an event in space in terms of Right
Ascension and Declination, together with some distance
measurement.
Any event is, therefore, fixed by recording its time
interval and its spaceinterval from some actual or hypo
thetical standard event. Now we have seen that both
these records vary with the world in which they are observed.
To say that the battles of Waterloo and Hastings occurred
at an interval of 749 years is an intelligible statement if
addressed to people on this Earth. It would convey no
67
68 READABLE RELATIVITY
meaning, or in fact a false one, to some one in a world moving
rapidly along the line joining Hastings and Waterloo. If
these two events had been recorded in this other world,
the timeinterval (and the spaceinterval) would be quite
different. It is obviously desirable to try to discover
some property connecting the two events whose measure
will have the same numerical value, in whatever world
records of the events are made. If this can be done, we
can regard such a property as something absolute, inde
pendent of all observers, retaining the same value however
it is viewed. Measures of timeintervals and distance
intervals are not absolute in this sense : they vary from
world to world. But nere is a kind of union of the two
about which opinions do not change, a union which will
be measured with equal magnitudes by all observers. We
shall first consider a numerical example to show what the
nature of this union is.
Records of the Interval between Two Events by Various
Observers.
A's world observes two special events and notes that
Event II. occurs 12 seconds later than Event I. and at a
distance 4 luxes due east of it. The points at which
Events I., II. occur will be called E, F respectively.
A writes his record of the interval between the events
in the form (4 ; 12). With our previous notation, this is
short for %=4 ; t x =i2.
(i) O is an observer who says that A's world is moving
away from him at f lux per second due east. How does
O record the interval between the events ?
We have the relations
x x +ut x t x \ux x
x V(z—u?)' l ~V(iu*)
where (x ; t) is O's record and x x =/\, ^=12, w=§.
THE SEPARATION OF EVENTS 69
V(i« 2 ) = v^(iA) = ^(M)=I
4+f X 12 _2o+36 _56
f 4 4
and , = Hdl2<_4 = 6o±H = e =l8
t 4 4
.*. O records the interval between the events as 14 luxes
and 18 seconds, or, more shortly, (14 ; 18).
(ii) P is an observer who says that A's world is moving
away from him at f lux per second due west. How does
P record the interval between the events ?
Using the same formulae as before, we now have
%=4> ^i= I 2, «=— f
... V(i^) = v/(i^ T ) = v / (if)=
. , 4+(f)*g _ gQ36 ^ = __ A
' ' — 4 A A
and , = H±tiL4 = 6o=i2 = l8 =I2
t 4 4
/. P records the interval as —4 luxes and 12 seconds, or,
more shortly, (—4 ; 12).
This means that P says Event II. occurs 4 luxes west
of Event I.
(iii) Q is an observer who says that A's world is moving
away from him at ± lux per second due east. What are
Q's records ?
Here %=4, t x =ia, w=
.. v(i w 2 HV(iM)=v(A)=f
. 4+(?)i2 ^ 2Q+48 _68
. . x — 3 3 3
i 3 3
.*. Q records the interval as
68, 76 .__„ /68. 76 s
3
68, 76 , ft
— luxes, '— seconds, or ( 
3 3 v
70
READABLE RELATIVITY
We will set down the records of three other observers,
leaving the calculations to the reader,
(iv) R is an observer for whom u=— f.
Show that R's record is ( ; — Y
V 3 3/
(v) S is an observer for whom w=i§.
Show that S's record is (■— ; ~\
(vi) T is an observer for whom w=— 2 r_.
Show that T's record is (  ; — ).
\3 3/
Let us now collect these results in a single table, arranging
them so that the distanceintervals are in ascending order
of magnitude, apart from sign.
A.
Observer.
T.
p.
R.
O.
Q
S.
—
Value of u luxes per second
A
_ S
4
12
_ 4
■5"
4
6
«
4
12
Distanceinterval, x luxes .
Timeinterval, t seconds .
2
s
9i
Ml
14
18
22f
25i
39i
4°!
This table shows that if in passing from one world to
another the distanceinterval (apart from sign) is increased,
then the timeinterval is also increased.
What relation connects the values of t and x ? A
glance at the table does not suggest any obvious relation.
But if we write down corresponding values of t 2 and x 2 ,
it is not difficult to guess the answer. This is done below :
Thus for R we have t=\^%, x=— 9 J ;
therefore 2 2 =(i4f) 2 =^X^=^=2i5i,
and^={ 9 i) 2 =xf=^=8 7 i.
THE SEPARATION OF EVENTS 71
T.
P.
R.
O.
6 4 i£
5i3i
S.
t 2
* 2
144
16
128I
*
144
16
2i5i
87*
324
196
1664^
I536H
f 2 * 2
128
128
128
128
128
128
128
In each case, we see that t 2 — % 2 =i28.
Although the timeinterval t seconds and the distance
interval x luxes varies from one world to another, all
observers alike agree that the value of t 2 — x 2 is 128.
We shall represent the expression t 2 — x 2 by s 2 , so that
in the special case above we have s 2 =i28 or s=V(i28)
=113, as measured by every observer : and we say that
n3 measures the separation of the two events. This
name is due to Professor Whitehead.
The separation of two events defined by the formula
S 2=*2_#2
is an entirely new conception ; it is neither time nor distance,
but some kind of fusion of the two. Its importance arises
from the fact that it is independent of the world in which
the records are made ; all observers attribute to it the same
numerical measure, provided it is agreed to measure time
in seconds and distance in luxes. The separation, therefore,
represents something absolute, some intrinsic property
connecting the two events, without regard to the conditions
under which the events are observed.
Formal Treatment of Separation.
If, with our previous notation, A records the interval
between two events as x 1 luxes, t x seconds, and if O records
the interval between the same events as x luxes, t seconds,
then we can prove that t 2 — x 2 =t x 2 — % 2 .
72 READABLE RELATIVITY
Using the formula x= *} +ui * • t= *}+ UXl
we have t*x*J^±^&l±^
l—U 2 l—U 2
_ (t 1 2 +2ut 1 x 1 \u 2 x 1 2 )  {xf+zut^+uH*)
I— u 2
_ t 1 2 +2Ut 1 X 1 +U 2 X 1 2 —X 1 2 —2Ut 1 X 1 —U% 2
l—U 2
= ^(I_ W 2)_ % 2( I _ W 2)
l^U 2
=t 2 x 2 .
If then we represent t 2 —x 2 by s 2 , we see that tf—xf
also equals s 2 , and therefore the value of s as calculated
by one observer is the same as the value of s calculated
by any other observer. Consequently the separation
between two events, as defined above, survives transforma
tion from one world to another. An expression of this
kind is called an invariant. Its measure has nothing to
do with the circumstances of the observer; it represents
an objective relation between, or a physical property of,
the events themselves.
Real and Imaginary Values of the Separation.
In the numerical example discussed above (p. 68), the
value of t 2 — x 2 was positive, and therefore the value of s,
its square root, was real. But if t is less than x, then
s 2 =/2_^2 ==a negative^number ; in this case s is the square
root of a negative number (as for example V~ Y6) and
is therefore imaginary. This does not of course mean
the events are imaginary, but merely that the measure of
the property, called the separation of the events, is ex
pressed in some cases by an imaginary number. It is
easy to see when this will happen.
Suppose Event I. occurs at E and Event II. occurs at
F, and that the interval is x luxes, t seconds. Then a ray
THE SEPARATION OF EVENTS 73
of light will take x seconds to move from E to F. Suppose
now the timeinterval between Events I. and II. is less
than the time a lightray takes to travel from E to F, then
t is less than * and therefore s 2 =t 2 — x 2 =a negative number.
Therefore the separation is imaginary if the timeinterval
between the events is less than the time required to send a
light signal from one place to the other : in other words, if
the separation is imaginary it is impossible for a message,
sent off from the place E where Event I. has occurred,
describing that event, to arrive at the place F where Event II .
occurs before the actual occurrence of Event II. This
fact will be recognised by all observers alike, for if one
observer obtains an imaginary valufe for the separation,
so will they all.
If, however, s is real, t is greater than x. Consequently,
if as soon as Event I. has occurred at E a wireless signal
reporting the event is dispatched to F, every one will agree
that it will reach F before Event II. occurs.
The intermediate case when s is zero is easy to interpret.
This requires that t=x or that the timeinterval between
the events is equal to the time a lightsignal takes to travel
from the one place to the other. Suppose that Event I.
is the dispatch of a lightsignal from E, and that Event II.
is the receipt of this lightsignal at F, then t=x and the
separation is zero. The Sun is 93,000,000 miles, or 500 luxes,
away from the Earth. If Event I. occurs on the Sun
and if Event II. occurs after a timeinterval of 8 minutes
20 seconds (=500 seconds) on the Earth, then the separa
tion between these two events is zero.
It is important to remember that when an observer puts
on record the time of an event, he does not give the time
at which he sees the event take place, but the corrected
time after allowing for the time the lightray has taken
to reach him — i.e. he records the actual time at which he
believes the event took place, not the time at which he
74 READABLE RELATIVITY
saw it happen. For example, the observed time of an
eclipse of one of Jupiter's moons would depend on how
far away Jupiter was from the Earth when the eclipse was
observed. To find the time at which the eclipse really
happened, allowance must be made for the time the light
ray took to reach the Earth. It was the discrepancy
between the calculated times and the observed times of
the eclipses of Jupiter's moons which first enabled a Danish
astronomer, Roemer, in 1675, to calculate the velocity of
light.
TimeOrder of Two Events.
Suppose that O records the interval between Event I.
at E and Event II. at F as x luxes, t seconds, where x and
t are each positive, so that Event II. occurs after Event I.
and distances in the direction E to F are measured as
positive. Suppose another observer A is moving relatively
to O in the direction E to F at u luxes per second, so that
m is also positive. A records the timeinterval as t x seconds.
Then we know that
t—ux
* 1_ V(iw 2 )
If t x is positive, A says that Event II. occurs after Event I. ;
but if t x is negative, A says that Event II. occurs before
Event I.
Is it conceivable that, after all proper corrections (as
noted above) have been made, O and A should disagree
as to the order in which the events took place ? The
answer is bound up with the nature of the separation
between the events.
(i) An Imaginary Separation.
If the separation is imaginary, t 2 is less than x 2 , and
therefore, as each is positive, t is less than x.
Put t=vx so that v<i.
THE SEPARATION OF EVENTS 75
m
' t—ux vx—ux _ x(v—u)
We thenjiave *i= ^^y^.^^.^
Now u can never be greater than 1 ; no observer can
move faster than a light ray. But u can have any value
between o and I, because we can think of A as moving
away from O with any velocity up to 1 lux per second.
.*. since v<i, we can think of an observer A who is
moving away from O with a velocity greater than
v luxes per second, i.e. so that u>v.
But if u>v, v— u is negative
.*. t x is negative
.'. this observer A says that Event II. occurs before Event I.
Again, since v<i, we can choose an observer B who is
moving away from O with a velocity exactly equal to v luxes
per second, i.e. so that u—v.
But if u—v, v—u=o
/. ^=0 ,
.\ this observer B says that Event II. occurs simul
taneously with Event I.
Again, we can obviously choose an observer C who is
moving away from O with a velocity less than v luxes per
second, i.e. so that u<v ; then v—u is positive
.*. t x is positive
/. this observer C agrees with O that Event II. occurs
after Event I.
Now all these observers are equally entitled to their own
views. Their timerecords are of course actual as opposed to
observed times — that is to say, the observer takes the time
of his clock when he sees the event happen, and corrects
for the time the lightray has taken to reach him. In this
way he obtains what he calls the actual time at which the
event took place. But as no preference can be given to
any one observer over any other, we are forced to conclude
76 „ READABLE RELATIVITY
that it is meaningless to attach a timeorder to events whose
separation is imaginary. For such events we may say,
using comparative terms, that whereas in slowmoving
worlds Event II. occurs after Event I., in fastmoving worlds
Event II. occurs before Event I., and there is one world
in which the Events are actually simultaneous. Another
descriptive form of this statement is based on the idea
that the occurrence of any event cannot be said to be
caused by any event which follows it. If some observers
say that Event I. occurs before Event II., while other
observers with equal justice say that Event II. occurs before
Event I., then it is impossible to imagine that there is any
causal relation between the events. Now this is what
happens when t is less than x, i.e. when the timeinterval
between the events is less than the time a lightray takes to
travel from one place to the other. If, then, we substitute
for " causal relation " the word " force," and if we say that
neither event can exert a force on the other event, we are
really saying that no force can be propagated with a
velocity greater than that of light. If, for example, the
Sun is regarded as exerting a gravitational force on the
Earth, the propagation of this force takes at least 500
seconds to reach the Earth from the Sun. We shall,
however, see later that Einstein's theory abolishes the idea
of a gravitational force altogether.
(ii) A Real Separation.
If the separation is real, t 2 is greater than x 2 , and there
fore t is greater than x.
Put t=vx so that v>i.
Then, as before, t,=~r, 4r.
1 v(i— u 2 )
Now u can never be greater than 1, but v is greater than 1 ;
.". v>u
.'. t x is always positive.
THE SEPARATION OF EVENTS 77
Every observer will therefore agree with O that Event II.
occurs after Event I.
Consequently, if the separation is real, the timeorder of
the two events is definite, the same for every one.
We know that different observers make different time
sections of the Universe : but all observers alike will place
Event I. in a timesection which precedes their timesection
containing Event II. For each of them, Event I. appears
as a feature in the story leading up to Event II., and may
therefore be said to contribute, however indirectly, to the
occurrence of Event II. We may say that when the
separation is real there is some causal connection between
the events.
Proper Time.
Suppose with our previous notation A records two events
which happen to him. He says the timeinterval is t x
seconds, and the disfenceinterval is zero, for both events
occur at the same place, namely, the place where he is.
The separation between the events is given by s i =t 1 2 — o 2
=t x 2 or s=<j.
Now consider an observer O who says that A's world is
moving away from him at u luxes per second : suppose
as usual that Event I. occurs at zero hour when A is passing
O. Then O records the interval between the events as
x luxes, t seconds. The following diagram exhibits the
two views :
A' s opinion
0' s opinion
Event I
Event II
ut,
X"
0,A
X
0,A
ut
1
O
•X
A
O
A
Fig. 21.
I
78 READABLE RELATIVITY
Since both events happen to A, says that the distance
interval x luxes equals ut luxes ; x—ut.
O also says the separation is given by s 2 =t 2 — x 2
:. t 1 2 =s 2 =t 2 —x 2 =t 2 —uH 2 =t 2 (i—u 2 )
.: t x =w{i—u 2 ).
Now t x seconds is the timeinterval recorded by A, the
person to whom the events happen. It is the time accord
ing to which each individual records the incidents of his
own life, and it is called the " Proper Time " for the indi
vidual concerned. Since V(i— u 2 ) is less than 1, we see
that t x is less than t, and consequently the proper time
between two events is less than the timeinterval recorded
by any other observer. If we consider the lifehistory of
an individual, the measure of the timeinterval between two
events will depend on the observer ; but if we regard the
individual as carrying his own clock about with him, we
call his measure of the timeinterval the " proper time," and
it is less than that of any one else. He is in fact measuring
the separation between two events wholly in time, while
other observers measure it. partly in time and partly in
distance. An observer whose measure takes more distance
must also automatically take more time as well, in com
pensation. The separation between two events is equal to
the proper time between them, that is, the timeinterval
as measured by the person to whom the events happen.
Of course if the separation is imaginary it is impossible for
the two events to happen to the same person, and " proper
time " ceases to exist.
Exercise vi
1. With the data on pp. 68, 70, verify that the records of
R, S, T are as given in the text.
2. With the data on p. 68, can you find an observer who will
say that the events occurred at the same place ? If so,
THE SEPARATION OF EVENTS 79
what velocity does this observer attribute to A, and
what is his measure of the timeinterval ?
3. A gives the interval between two events in the form #1=3,
*i = 5 O says that for A's world u=^g. What are O's
records for the interval ? What is the separation ?
4. A records the interval between two events as 5 luxes,
13 seconds; O records the timeinterval as 15 seconds.
What is O's record of the distanceinterval ? What
is the separation ? What velocity does O attribute
to A ?
5 . O says that Event I. at E is given by * = 2, x = 5 , and Event II.
at F is given hy t' —6, x' = 12 ; units being seconds and
luxes. What are the time and distanceintervals as
recorded by O ? If A is moving away from O in the
direction E to F at $ lux per second, what is the time
interval according to A ? Interpret your result. What
is the separation ?
6. O says that two events happen at different places at the
same time. What can you say about their separation
and their timeorder ?
7. O says that two events happen at different times at the
same place. What can you say about their separation
and their timeorder ?
8. With the data of No. 5, can you find an observer for whom
the events will be simultaneous ?
9. On the same day the following events are noted :
Event I. Earthquake at Tokio at 12 noon.
Event II. Formation of a Sunspot at 12*06 p.m.
Event III. Disappearance of the Sunspot at 12.12 p.m.
What can you say about the timeorder of these events ?
[O. I get out of bed at 7 a.m. and retire to bed at 10 p.m.,
Greenwich time ; an observer says that my bed has
moved 72,000 luxes in the interval. How long does he
say I have been out of bed ? If I have moved about
with the velocity of light all the time I have been out of
bed, how long do I say the time is ?
CHAPTER VII
THE FOURTH DIMENSION
" There is no difference between Time and any of the three
dimensions of space except that our consciousness moves along
it. . . . The civilised man can go up against gravitation in a
balloon ; why should he not hope that ultimately he may be
able to stop or accelerate his drift along the Time dimension,
or even turn about and travel the other way ? " — H. G. Wells,
The Time Machine.
Hitherto we have only considered events which occur at
points in a straight line along which the worlds are separ
ating. It is easy to extend what has been said so as to
include events occurring at any points in space.
Points in a Plane.
If we take two perpendicular lines in the plane, we can
fix the position of any point in the plane by stating its
distances from these
two lines. For ex
ample, take a point O
and draw from it a
line Ox due east and
a line Oy due north.
Suppose any point A
is 5 miles east and 3 miles north of O ; this fixes the position
of A. If we start at O and walk 5 miles #wards, i.e.
eastwards to M, and then walk 3 miles jywards, i.e. north
awards along MA, we arrive at A. The distances 5 and 3
are called the coordinates of A ; we say that the point A
is given by #=5, y=% or that its coordinates are (5, 3).
6
W
O 5 m x
Fig. 22.
82 READABLE RELATIVITY
The #coordinate is always put first. Suppose that B is
another point in the plane and that its coordinates are
(9, 6), then we can move from A to B by going a farther
distance 9—5=4 miles awards and 6—3=3 miles ywards.
In the figure, AK=4, KB =3
/. AB 2 =AK 2 +BK 2 =4 2 +3 2 .
In the same way, if the distanceinterval between A and B
is X units awards and Y units ^wards, so that AK=X and
BK=Y, we have
AB 2 =X 2 +Y 2 .
The coordinates of a point are simply its distance
intervals from O measured (i) awards, (ii) ^ywards. If we
know the coordinates of any two points A and B, we can
find the distanceintervals of B from A measured awards
and ^ywards by subtracting the #coordinates and the
jycoordinates, as above.
Suppose now an observer O says that the world of A, B
is moving away from him in the direction Ox at u luxes per
second, and that an Event (I.) occurs at A and another
Event (II.) occurs at B after an interval T seconds. Then
O says that the interval between the events is T seconds,
X luxes awards, Y luxes ywards. Now in the world of A,
the timeinterval is different, say T x seconds ; and the
^interval is different, say X 2 luxes ; but the ^/interval,
say Y x luxes, is the same, because the rules of O and A
agree when across the stream, therefore Y=Y r
Now we have already proved that T 2 — X 2 =T X 2 — Xj 2
/. since Y=Y 1 , T 2 X 2 Y 2 =T 1 2 X 1 2 Y 1 2 .
But if O measures the length of AB as r luxes and A
measures the length of AB as r x luxes, we know that
X 2 +Y 2 =r 2 and X 1 2 +Y 1 2 =f 1 2 .
.. T 2 f 2 =T 1 2 r 1 2 .
We therefore put s 2 =T 2 X 2 Y 2 =T 2 f 2 , and call s
the separation between the events whose intervals are
THE FOURTH DIMENSION
83
(X, Y ; T) according to O and (X lf Y x ; T^ according to A,
and whose spaceintervals are r, r x according to O, A.
Points in Space./
To fix the position of any point in space, we take
three mutually perpendicular planes, called the planes of
reference, and measure the distance of the point from
each plane.
Suppose, for example, we draw on a horizontal plane a
line Ox due east, a line Ov due south, and a line Oz vertically
upwards. A point A in the air could be fixed by saying it
was 4 miles east of O, 3 miles south of 0, and at a height of
2 miles above O. If we start at O and walk 4 miles awards,
i.e. eastwards to P, and then walk 3 miles .ywards, i.e. south
wards along PM to M, and then rise 2 miles 2wards, i.e.
vertically upwards along MA, we arrive at A . The distances
3, 4, 2 are called the coordinates of A ; we say that the
point A is given by x =4, y=3, z—2, or that its coordinates
are (4, 3, 2). The coordinates are always written in this
order, x, y, z.
Fig. 23.
Now suppose we move from A to any point B by travel
ling X units #wards along AK to K, and then Y units
84 READABLE RELATIVITY
^wards along KH to H, and then Z units 2wards along
HB to B. We say that the distanceintervals of B from A
are X, Y, Z.
Now ZAKH is a right angle,
/. AH 2 =AK 2 +KH 2 =X 2 +Y 2 .
Also ZAHB is a right angle,
.'. AB 2 =AH 2 +HB 2 =X 2 +Y 2 +Z 2 .
This shows how the length of AB can be calculated as
soon as we know the distanceintervals of B from A, and as
before we can calculate these, by subtraction, if we know
the coordinates of A and B.
Suppose now an observer O says that the world of A, B
is moving away from him in the direction Ox at u luxes per
second, and that an event (I.) occurs at A and another event
(II.) occurs at B after an interval T seconds.
Then O says that the interval between the events is
T seconds, X luxes %wards, Y luxes ^wards, Z luxes
2wards. Suppose A says that these intervals are respec
tively T x seconds, X lf Y x , Z x luxes. Then we know that
although T and X are not equal to T x and X lt yet
T 2 — X 2 =T! 2 — X x 2 .
Further, Y=Yj and Z=Z X , because the rules of A and O
agree when put across the stream.
/. T 2 — X 2 — Y 2 — Z 2 =T! 2 — Xi 2 — Yi 2 — Zj 2 .
Suppose O and A measure the length of AB as r luxes and
r x luxes respectively.
Then r»=X 2 +Y 2 +Z 2 and r 1 2 =X 1 2 +Y 1 2 +Z 1 2
/. T 2 — r 2 =T 1 2 — rf.
We therefore put s 2 =T 2 X 2 Y 2 Z 2 =T 2 r 2 , and call
s the separation between the events whose intervals are
(X, Y, Z ; T) according to O and (X^ Yj, Z x ; Tj) according
to A, and whose spaceintervals are r, r x according to O, A.
We have now applied the idea of separation to any two
THE FOURTH DIMENSION 85
events occurring at any points of space and time, and have
thereby constructed a function or expression which may
properly be called a physical reality.
FourDimensional SpaceTime.
In order to specify the time and spaceintervals between
two events, we see that it is necessary to give values to
four different variables, each of which can change inde
pendently of the others, namely X, Y, Z, T.
Although each observer distinguishes sharply between
space and time, the distinction drawn by one observer is
not the same as that drawn by another. What one
measures as " time," another measures partly in space
and partly in time. The distinction, therefore, between
space and time is subjective — that is to say, the observer,
although unconsciously, is affected by his circumstances in
his discrimination between the two. We cannot therefore
suppose that this distinction corresponds to an objective
physical reality. And so we are forced to the conclusion
that we live in a fourdimensional world, divided arbitrarily
by each observer into three dimensions of space and one
dimension of time, but in reality an entity to be called
spacetime.
Owing to the fact that people on the Earth never move
relatively to each other with really high velocities, the time
and distance axes of any one person agree fairly well with
those of any other. Life would be a perplexing and em
barrassing matter if this agreement did not exist. But
the distinction between space and time drawn by an in
habitant of a Beta particle moving with a velocity approach
ing that of light is so widely different from that of the
physicist observing him that any kind of common social
life would be unthinkable.
Every observer maps out the Universe with his own
space and time axes. Suppose an observer A were to
86 READABLE RELATIVITY
classify a large number of events all of which occurred at
the same moment. Then this collection of events forms a
timesection or timecleavage by A of the Universe. Now
events which are simultaneous for A are usually not simul
taneous for O. Suppose O takes one event in A's list and
proceeds to make a catalogue of events simultaneous with
it. Then we know that O's list will not agree with A's list :
in other words, O's timecleavage of the Universe is not the
same as A's ; they may have some members in common,
but speaking in general terms there will be far more dis
agreement than agreement.
The Universe is to be regarded as a collection of events,
anywhere and anywhen, an entity which mathematicians
call a continuum, and the difference between O and A is
simply that they slice it up differently. The Universe as
an entity is timeless (and spaceless). What each indi
vidual perceives is merely his own timesection. History
records some of the timesections of our ancestors, and
Mr. H. G. Wells forecasts timesections of our descendants.
With neither group have we the power to obtain direct
acquaintance, merely because we cannot put ourselves
into the position in which the desired timecleavage would
be the natural one. But all events, past, present, and
future as we call them, are present in our fourdimensional
spacetime continuum, a universe without past or present,
as static as a pile of films which can be formed into a reel
for the cinematograph. It is obviously absurd to attempt
to form a picture of a fourdimensional Universe, but it
may suggest ideas if we consider the structure of a three
dimensional Universe whose inhabitants partition it by
two axes of space and one axis of time.
FlatLand.
Suppose a worm can crawl anywhere on a vast horizontal
plane surface, but has no power of raising itself out of the
THE FOURTH DIMENSION
87
>x
Fig. 24.
plane or burrowing into it ; and, better still, imagine that
the worm has no idea that there is even such a thing as
" above " or " below." In that case the worm is living in
a threedimensional con
tinuum, two dimensions
in space and one dimen
sion in time, forming his
spacetime world.
Draw two perpen
dicular straight lines ox,
oy on the plane, and draw
a. line ot perpendicular to
the plane to represent the
timeaxis. The worm, of
course, cannot picture this
line ot any more than we
can visualise a fourth axis of reference in addition to three
mutually perpendicular axes chosen for our spaceaxes.
Suppose the worm starts its life at the point a, co
ordinates x x , y x , at a time an observer O at rest at records
as t t seconds past zero. Then O denotes this event by the
point A in space whose coordinates are x v y lt t v As the
worm moves about on the plane, each event in its life is
represented by a point in space. Finally, the worm dies
at b, coordinates x 2 , y 2 , at a time O records as t z seconds
past zero. Then O represents this event, the worm's
death, by the point B in space, coordinates x i} y 2 , h The
whole history of the worm from birth to death is then
represented by a curve in space starting at A and ending
at B. And we say, using Minkowski's phrase, that the
curve AB is the " worldline " of the worm. Suppose now
that there are numerous worms and other objects in the
plane. Each has its own worldline, and if we take the
complete collection of all these worldlines, they constitute
the spacetime worm Universe. The collision of any two
88 READABLE RELATIVITY
objects, in fact the happening of any event, is recorded by *
an intersection of two worldlines. If O compiles a cata,
logue of simultaneous events, he is simply making a time
section of these worldlines, and is choosing points on the
worldlines which are at the same height above the plane
xoy.
Worms who move slowly will agree with O's conclusions.
But a rapidly moving worm R will make such different
time and space measurements that the worldlines he
constructs will (so to speak) be quite a different shape from
O's. The shape, however, hardly matters. If two of O's
worldlines intersect, the corresponding worldlines of E
must also intersect, because any intersection represents a
spacetime event. Consequently, if we think of O's world
lines as forming a vast network in space, then R's world
lines will also form a network. The meshes in the one
system may be quite different in size and shape from those
in the other ; but to each mesh and each corner of that
mesh of O's network there will correspond a unique mesh
and a unique corner of that mesh of R's network.
Straight WorldLines.
Worldlines can clearly be of any shape ; given the
observer, the shape is an index of the worm's history.
What meaning must be attributed to the statement that a
worldline is straight ?
Choose the origin O so that the worm is born at O at
zero hour and dies at P, coordinates X lf Y x at T 2 seconds
past zero according to the observer O. Draw PC parallel
to the timeaxis and of length T x units ; draw PM per
pendicular to Ox so that OM=X x , MP=Y r
Then O and C represent in spacetime the birth and
death of the worm. Suppose the worm's worldline is
the straight line OC. Let K, any point on OC, repre
sent an event in the life of the worm, and suppose the
THE FOURTH DIMENSION
89
coordinates of K are X, Y, T; in our figure, ON=X,
NQ=Y, QK=T.
Fig. 25.
, KQ OQ QN
From similar triangles, rp = OP == pM :
Our coordinates measure the distance and timeintervals
since the birth of the worm.
ON
: OM
' * T 1 "~OP~Y 1 Xj
This means that the timeintervals and the spaceintervals,
measured from the birth of the worm, all increase at the
same uniform rate ; and it also follows that the separation
of an event from O increases at the same uniform rate.
The observer, therefore, says that a worm, whose world
line is straight, is moving with uniform speed in a straight
line : on the Newtonian Theory, the worm would be
moving freely, uninfluenced by any force.
A numerical example may help to make these ideas
more intelligible.
go
READABLE RELATIVITY
Let us suppose that the progress of the worm is repre
sented by the following table, the units of time and space
being seconds and luxes :
Event.
A.
o
B.
13
C.
26
D.
39
E.
52
F.
65
G.
78
H.
91
K.
104
TimeT
^coordinate X
o
3
6
9
12
15
18
21
24
ycoordinate Y
o
4
8
12
16
20
24
28
32
In this table the space and timeintervals measured
from the start all increase at the same uniform rate.
Take, for example, the interval between the events
D and E ;
T=52— 39 =I 3 ; X=i2— 9=3 ; Y=i6— 12=4
.*. the spaceinterval of E from D= V(X 2 +Y 2 ) = ^(9+16)
the separation of E fromD = V(T 2 — X 2  Y 2 ) = /(I69— 25)
= v / i44= I2 .
We should obtain the same numerical results for any
other pair of successive events.
Further, the spaceinterval of K from A is the sum of
the spaceintervals A to B, B to C, C to D, . . . H to K ;
and the separation of K from A is the sum of the separa
tions B from A, C from B, D from C, . . . K from H.
For the spaceinterval of each portion is 5, and there
fore the sum of the eight spaceintervals is 8x5=40 ;
while the space of interval of K from A is
v / 24 2 + 322=^576 + 1024= ^1600=40. '
Similarly, the separation of each portion is 12, and there
fore the sum of the separation for the eight intervals is
12x8=96 ; while the separation of K from A is
Vio4 a — 24 2 — 32 s = ^10816— 1600 = ^9216 =96.
THE FOURTH DIMENSION 91
Curved WorldLines.
The fact that if a line is straight, the distance of the
endpoint from the startingpoint is equal to the sum of
the lengths of the various portions of the line is a funda
mental fact in ordinary geometry. We have now seen that,
correspondingly, if a worldline is straight the separation
of the last event from the first event is equal to the sum
of the separations between successive events measured
along the line.
If a line is curved , the distance of the endpoint from the
startingpoint is less than the sum of the lengths of the
various portions of the curved line. In contrast to this,
we shall show that if a worldline is curved , the separation
of the last event from the first event is greater than the
sum of the separations between successive events measured
along the worldline.
A worldline is associated with the history of an actual
particle ; it represents the movement of the particle as
viewed by some observer. The speed of the particle
cannot, therefore, exceed the velocity of light. If the
interval between any two events in the history of the
particle is measured by T seconds, R luxes, we know that
T> R and therefore the separation is real. In the numerical
example, given above, if T=i3 we have R=5
Now consider a worm whose
worldline consists of two straight
portions AB and BC. Suppose the
events A, B, C are recorded by O as
follows :
A, t=o, x=o, v=o ;
B, 1=13, x=2, v=5 ;
C, *= 2 6, *=6, v=8 ; FlG  26 
the units being seconds and luxes.
92 READABLE RELATIVITY
Then the intervals are
A to B, ^=13, X x =2, Y lS =5
B to C, T a =26i3=i3, X 2 =62=4, Y 2 =85=3
••T 1 " I 'X i T*»Y 1 »"
.*. the coordinates do not increase at equal rates, and so
the worldline ABC is not straight.
The separation of B from A is
■v/( I3 2_ 2 2_ 5 2) = v / (l69 _ 4 _ 25 ) = V(I40) .
The separation of C from B is
a/(i3 2 4 2 3 2 ) = ^(169169) = ^(144) =12.
The separation of C from A is
</(26 2 6 2 8 2 ) = ^(6763664) = V (576) =24.
But \/(i4o) is less than 12
.*. the separation of B from A+the separation of C from
B is less than the separation of C from A.
We see, then, in this case that if the worldline ABC is
not straight, tfee separation of C from A is greater than
what we may call the separation measured along the world
line ABC.
We shall now give a general proof of this result.
Straight and Curved WorldLines.
If the worldline ABC of a particle is straight, we know
that it moves with uniform velocity in a straight line.
This involves two things :
Suppose the time and spaceintervals from A to B are
T x seconds, r x luxes, and the corresponding intervals from
B to C are T 2 seconds, r 2 luxes.
Then (i) the spaceinterval of C from A is 7i+f 2 luxes.
(ii) ^r =q^, for these fractions represent the speed
of the body from A to B and from B to C.
THE FOURTH DIMENSION 93
It is, of course, true that the timeinterval of C from A
is T x +T 2 seconds, but this is true whether the worldline
is straight or curved, if we are dealing with the progress
of a body.
Now if either of the conditions (i) and (ii) ceases to be
true, the worldline is no longer straight ; the portion AB
will no longer be in the same straight line with BC.
Condition (i) holds only if the spacemovement of the body
is a straight line ; condition (ii) holds only if the body
moves at a constant speed. Each condition is necessary
for uniform velocity. Thus if the body describes a straight
line at variable speed, (i) holds and (ii) fails, and if the
body describes (say) two sides of a triangle in the plane
with constant speed, (i) fails and (ii) holds. In each case
the worldline is not straight. Of course if the body
moves in a plane curve with variable speed, both con
ditions fail.
Separation measured along a WorldLiije.
If the worldline AB of a body is straight, with the
above notation, we know that the separation of B from
A is VCIV*— R x 2 ). We can represent
this geometrically by drawing a right
angled triangle abH, so that ab=T 1 ,
aH=Rj, and AaRb=go°.
Then by Pythagoras, &H 2 +R 1 2 =T 1 2 ,
or fiH^T^Rx 2 ,
.. 6H=V(T 1 2 R 1 2 ).
The side 6H therefore measures the separation.
Now suppose the worldline AC of a body is curved. Take
a large number of successive events in the life of the body,
D, E, F, G, . . . N. If we take a sufficient number, each
portion AD, DE, EF, . . . of the worldline is nearly
straight, and we can then find the separation, as above,
94
READABLE RELATIVITY
for each of these portions. If we add together all the
separations for the small portions, we say that the sum
is the separation of C from A measured along the worldline.
We shall prove that this is less
than the separation of C from A.
To do this, it will be sufficient to
show that if AB and BC are two
different straight worldlines, then
the separation of C from B+the
separation of B from A is less
than the separation of C from
A. For we can then argue as
follows :
The separation CA is less than the
sum of the separations CD, DA ;
which is less than the sum of the separations CE, ED, DA ;
which is less than the sum of the separations CF, FE, ED*
DA;
and so on.
Fig. 28.
Maximum Separation.
Suppose the worldline of a body consists of two different
straight portions AB, BC, and suppose the time and space
intervals of B from A are T x seconds, r x luxes, and that
the corresponding intervals of C from B are T 2 seconds,
r 2 luxes. Suppose also that the corresponding intervals
of C from A are T seconds, r luxes. Then T=T!+T 2 ;
and r cannot be greater than r x +r 2 . Since AB is not in
a straight line with BC, we know that either ^ is unequal
to j or that r x +r 2 is greater than r. Possibly both these
conditions hold, but at least one must be true.
We shall represent the separations of B from A and
C from B by the method shown above.
THE FOURTH DIMENSION
95
Draw triangles aHb, 6KC with &K parallel to «H, and
so that aR=r lt ab=T v Z.aUb =90°
and bK=r 2 , bc=T 2 , £bKc=go°.
Produce cK and «H to cut at N, then 6KNH is a rectangle.
The separations of C from B and B from A are represented
by cK and 6H, but 6H=KN
.*. the separation of C from Bf the separation of B from A
=cK+&H=cK+KN=cN.
Now the separation of C from A= \/[T 2 — r 2 ]
= ^[(T 1 +T 2 ) 2 r>]
Now if t^ is not equal to ~r, abH. and bcK are triangles of
different shapes, so that the angles ball, cbK are unequal :
consequently ab is not in line with be
.*. ab\bc is greater than ac or T 1 +T 2 >oc.
Also r cannot be greater than r x +r t and r 1 +r a =aH+&K
=«H+HN=«N
.*. the separation of C from A =\ / [{T 1 +T i ) 2 —r 2 ]
>A/[ac 2 (f 1 +y 8 ) 2 ] or V[ac*aW]
>cN since ac 2 =aN 2 +Nc 2
> separation of C from B+separation of B
from A.
96 READABLE RELATIVITY
But if ^ should equal ~^, so that ab is in line with be,
then yj+^2 must be greater than r.
Now if ab is in line with be, ac=ab\bc=T 1 +T 2
.*. the separation of C from A
= ^[(Ti+Ta) 2 ^] = V[ac*r*]
> A /[flc 2 (/' 1 +f 2 ) 2 ] or V[ac*aW\
>cN as before
>separation of C from B+separation of B from A.
Consequently, if a worldline from A to C is not straight,
the separation of C from A measured along the worldline
is less than the direct separation of C from A — that is to
say, is less than the separation measured along the straight
worldline joining A and C.
The Geodesic Law of Motion.
Newton's first Law of Motion states that the progress
of a body under the action of no forces is represented by a
straight worldline. We can now state this law in more
general terms.
If a body is moving freely, and if A and C are two events
in its history, the spacetime path followed by the body between
A and C is such that the separation of C from A measured
along that path is a maximum.
The striking feature of this statement is thatrftj specifies
a path which is unique, and is independent of the axes of
reference or what is the same thing of the observers. There
are an unlimited number of paths joining two points in
spacetime, but there is one of these which stands apart
from all the rest in virtue of the fact that all observers
alike agree that it yields a separation greater than the
separation obtained by measurement along any other
path. The worldline which has this unique property is
called a geodesic.
It is interesting to note that if the body were to follow
THE FOURTH DIMENSION 97
a curved line from A to C travelling at the same speed as
a lightray, the separation along each portion of the curve
would be zero, since T=r for a lightray, and therefore
the total separation measured along this curved worldline
would also be zero. We can, in fact, unite C with A by
a curved worldline along which the separation can have
any assigned value from the maximum down to zero, but
the geodesic path is the only possible path for a body
moving freely.
The only objective property we have so far discovered
is this property called " separation." Time and space by
themselves merely express relations between the observer
and the thing observed : separation is the physical reality.
Now in the previous chapter we saw that the separation
between two events in the life of a body is equal to the
"proper time " for that body — that is, the timeinterval
measured by a clock which the body carries about. We
may therefore express the conclusions of this chapter by
saying that bodies in the Universe, if left to themselves,
follow the path which makes the proper time between their
birth and death as great as possible. A body chooses the
path which gives (in its own view) greatest length of life.
If A and B represent any two events in the life of the
body, the path from A to B is so chosen that its passage
occupies tike maximum amount of time (according to its
own clock).
This rule of conduct has been called by Mr.. Bertrand
Russell the " Law of Cosmic Laziness " ; it is the substitute
Relativity has made for the inclination of a body to follow
" the line of least resistance."
EXERCISE VII
1. An event A is given by # — 1, y=3, £=8, 2=35, and an
event B is given by x =4, y = 7, z = 20, t = 1 20, units being
luxes and seconds. What is the separation of B from A ?
7
98 READABLE RELATIVITY
2. With the data of No. i, the world line of a particle is the
straight line AB. What are the space coordinates of an
event happening to the particle when ^=52? What are
the coordinates of the event happening to the particle
for £ = 14 ?
3. With the data of No. 1, another observer says that the
events A and B both occur at the same place. What
timeinterval does he attribute to the two events ?
What is the nature of a straight world line joining A
and B, as judged by this observer, and how is his time
reckoning described ?
4. What is the general appearance of the world lines of
(i) Trafalgar Square Tube Station, (ii) a train on the
Bakerloo Tube, as observed by (a) a porter on the
platform, (6) the enginedriver, (c) a person in the Sun.
5. The position of a particle in space time is given by #=27**,
y = 36a, z=6oa, t = &$a, and the history of the particle is
obtained by giving a successively all values from o to 4.
What can you say about its worldline ? What is the
separation between the events corresponding to a =4 and
a=o ?
6. Three events A, B, C in the life of a particle which moves
in a plane are given by t=o, x=o, y=o ; t = i?, #=6,
y = i5; 2 = 34, at = 18, y =24. What is (i) the separation
of C from A, (ii) the sum of the separations of C from B
and B from A. Is the particle moving freely ?
CHAPTER VIII
MASS AND MOMENTUM
"We recognise certain varying states or conditions of matter,
and give one state one name and another another as though it
were a man or a dog. Of matter in its ultimate essence and
apart from motion we know nothing whatever. As far as we
are concerned, there is no such thing." — Samuel Butler,
Notebooks.
Composition of Velocity.
Suppose that in the world of A, B a body C starts from A
at zero hour (both by A's clock and O's clock) and moves
at a uniform rate to B, and
according to A travels the
distance AB=# 1 luxes in t x
seconds. Suppose A measures
the velocity of C as v luxes per
second.
■*>u
v
>
Then «=3. °>A B
** Fig. 30.
Suppose also that O says
that the world of A, B is moving in the direction A>B at
u luxes per second. What velocity does O assign to C ?
O says that the journey to B occupies t seconds and is of
length x luxes where (see p. 63)
*i+^i , . t 1 \ux 1
x= 7 =r and /= , 4,
VlM 2 VI— u 2
ioo READABLE RELATIVITY
:. says the velocity of C is j luxes per second where
x_ x 1 \ut 1 _ t x ' __ v\u
t~ U\ux^~~ , UXi~T\UV
1+7*
We may state this result as follows :
If a body C is moving at v luxes per second in the world
of A along AB, and if the world of A is moving away from O
in the direction A>B at u luxes per second, the velocity of
C as measured by O is — r — luxes per second.
This result disagrees with the Newtonian method of
composition of velocities. If a man is in a train travelling
at 20 feet per second, and if he throws a bottle forward in
the direction of motion of the train at 10 feet per second, we
should expect a workman on the line to say that the bottle
started moving at 20+10=30 feet per second. The for
mula just proved shows that it is not strictly accurate
to combine the velocities by simple addition. But of
course, for such small velocities as these, the correction is
inappreciable . Approximately 1 foot per second = — 9 luxes
per second
2 1
.*. the two velocities are — § and g luxes per second
— +—
io 8 io 8
.'.the composite velocity is luxes per second.
1 2
I+ io"
3_
10 8
X io 9 feet per second.
2_
I rr 16
IO 1
30
feet per second.
1000,000,000,000,000,2
which is indistinguishable from 30 feet per second.
MASS AND MOMENTUM 101
But for particles moving at very high speeds, such as,
for example, fi particles emitted by radioactive substances,
which can move with velocities as high as 099 luxes per
second, the correction introduced by this formula is
considerable.
It is interesting to take the extreme case where C is moving
with the velocity of light, i.e. v—t.
The Newtonian law of composition would make O assess
the velocity of C at i+m luxes per second. But the Rela
tivity formula gives x~=i lux per second, so that O's
measure is identical with A's measure. This result is of
course only a repetition of the statement which lies at the
root of the whole theory — all observers alike who measure
the velocity of light must obtain the same result.
Transverse Velocity.
Suppose, next, the body C starts from A at zero hour, and
moves at a uniform rate at right angles to AB at w luxes per
second. What velocity does O assign to C ?
Suppose A says that C travels y t luxes in t x seconds.
y,
Then w="~.
h
O says that C travels y luxes in t seconds, where
y=y t and *=v=^
But x 1 =o, since A says that C moves at right angles to
AB
V
O says that C moves with a velocity composed of
m>Vi_ m 2 luxes per second at right angles to AB, and
u luxes per second along AB.
102 READABLE RELATIVITY
Lastly, suppose that A assigns to C a velocity of v luxes
per second along AB, and w luxes per second at right angles
to AB. After t x seconds by A's clock, C has moved x x luxes
in the direction AB and y x luxes in the direction perpendic
ular to AB, where v=~ and w=^.
t i h
O says that t seconds have elapsed, and that C has moved
x luxes and y luxes in these two directions, where
x,\uL tAux x
%
:. O says that C's velocity along AB is j luxes per
second, where as before
x_ x 1 +ut 1 __ v\u
t~t 1 {ux 1 ~i\uv
y
and that C's velocity perpendicular to AB is j luxes
per second, where
y U\ux x , v,
y±
= V 1 —u 2 X — = Vi— w 2 X— j —
ux, i\uv
H
.'. C's velocity perpendicular to AB is
i luxes per second.
i\uv r
We see, therefore, that O's measure of C's transverse
velocity depends on C's longitudinal velocity.
An interesting illustration of this law of composition of
velocities is furnished by the motion of light through a
moving medium.
MASS AND MOMENTUM 103
FresnePs Convection Coefficient.
The velocity of light depends on the medium through
which it is propagated. In a vacuum it moves at 1 lux per
second, and through air its velocity is nearly as great ; but
for other media there may be considerable reductions. If
the refractive index of a medium is /u,, the velocity of light
through this medium is  luxes per second ; //. of course is
always greater than 1, for water it is about 133.
Suppose a ray of light is transmitted through a stream
of water, refractive index fx, which is itself travelling
through a tube at u luxes per second in the direction of
motion of the lightray. With what velocity relative to
the tube will the lightray advance ?
If we regard the ether as stationary, we should expect
the motion of the water to make no difference to the speed of
advance of the lightray. Its velocity in this case would be
 luxes per second.
/*
If we regard the water as carrying the ether with it, the
Newtonian law of composition of velocities would suggest
that the rate of advance relative to the tube would be
— \u luxes per second.
Experiments by Fizeau in 1851 and by Hoek in 1868
showed that the actual rate of advance lay between these
limits and amounted to +w(i s ) luxes per second. It
was suggested by Fresnel that this was due to an ether
drag ; that, in other words, the ether was only partly
carried along by the waterstream.
The Theory of Relativity supplies an alternative explana
tion. The lightray is moving at  luxes per second through
the water, which is itself advancing at u luxes per second.
104 READABLE RELATIVITY
Therefore an outside observer, using the law of addition
on p. ioo, says that the velocity of the lightray is
— \u
" luxes per second.
i +
H
Now
1 i ( T i V u \ x , u u 2
+« \+u)ii — ~) +u — 2
£ _ > /V  fJ,/ ft ' [A? /J.
I s
But u, the velocity of the stream of water in luxes per
second, is very small. Consequently we shall obtain a
good approximation if we neglect u 2 .
In this case, the velocity becomes
1 i u I , ( i\,
u'*' u ~u*~u'T u \ L 2/ luxes P er second.
This agrees with the expression required by experiment.
We therefore see that the experimental result is a close
approximation to that deduced by using the Relativity
formula. The expression i  2 is called Fresnel's Con
vection Coefficient, retails of Fizeau's and Hoek's ex
periments will be found in any standard textbook on Light.
Mass. •
In the Newtonian System of Mechanics we assign to
each body a number which measures a property of the
body called its mass. The mass of a given body is regarded
as a fixed thing, which is independent of its position or
velocity, or indeed of any influence brought to bear on it
so long as no part of the body disappears. Newton said
that the mass of a body was the quantity of matter it con
tained : this phrase is not a definition, but it serves to
suggest th 3 nature of the concept which the word represents.
MASS AND MOMENTUM 105
Newton's second law of motion states that the rate of
change of "the quantity of motion" of a body is pro
portional to the impressed force, and this may be said to
define either his conception of mass or his measure of force.
From that law we see that if the " quantity of motion "
of a body is defined by the product of its mass and its
velocity, the increase in the quantity of motion per unit
time is the measure of the force acting on the body.
Instead of the phrase "quantity of motion," the term
"momentum" is generally employed:
momentum =mass X velocity.
This conception of momentum plays a very important
part in Newtonian mechanics. If any number of bodies
are in motion, and if there are no external forces acting
upon the system, then the total momentum of the system
remains constant. The bodies in the system may exert
forces on each other, as, for example, by colliding ; but
such forces are not external and do not affect the sumtotal
of the momentum of the whole system. As the result of
collisions, etc., there may be a transference of momentum
from one member to another : the momentum which one
body gives away is received by one or more other bodies
of the system. But th£ quantity ot motion or momentum
of the system taken as a whole never varies unless some
external force acts on the system and in this way con
tributes momentum to the system. Momentum is all
asset for which an account can be kept. If a number of
individuals each possess so much money and if this system
of individuals neither receives money from outside nor pays
away money to outsiders, but merely engages in internal
financial transactions, the total capital of the system re
mains constant ; any individual member of the society
can only increase his store of money at the expense of one
or more other members. It may help the reader if we
press this analogy still further.
io6 READABLE RELATIVITY
We are comparing the total capital of a selfcontained
society and the number of members of that society with
the total quantity of motion (or momentum) of a self
contained system of bodies and the number of bodies in
the system. An observer who looks at the society may
have a different standard of the value of money from that
of its members. But his estimate of the total value of
the capital of the society will remain the same, whatever
interchanges of cash there are between the individual
members. His total estimate may not agree with that
made by another observer who has a different standard of
values, but, whatever it is, it will remain the same as long
as no new money is introduced into the system and no
money is abstracted from the system. Again, the observer
may see double : in that case he will, when counting the
number of individuals in the society, obtain a different
figure from that of an observer with normal vision. But
as long as there are no births nor deaths, no emigrations
or immigrations, his census figure will remain the same,
however the members of the society behave. The observer
is, in fact, applying a numerical measure to two properties
of the society, namely, the capital the society possesses
and the number of members of the society. Both of these
things are part of the nature of the society, independent
of any observer, although different observers may apply
different systems of measurement. But under the condi
tions we have enumerated we may say that there is a
Conservation of Capital and a Conservation of Membership
of the Society.
So when dealing with any number of bodies, forming a
selfcontained system, and not subject to any external
force, we shall say that there is a Conservation of Momentum
and a Conservation of Mass. The estimate of each of these
may vary from one observer to another, but whatever
internal forces may be at work in the system causing
MASS AND MOMENTUM
107
redistribution of momentum or mass, we say that the esti
mates of the total momentum and the total mass made
by any particular observer will remain constant.
Now let us suppose that the world of A, B contains two
bodies of masses m x , m 2 , moving with velocities v lt v 2 luxes
per second in the direction A>B, and let us suppose that
an observer O says that A's world is moving away from
him in the direction A^B at u luxes per second.
Fig. 31.
A says that the total mass of the system is
m 1 {m 2 =c, say,
and the total momentum is m 1 v 1 {m z v 2 =d, say.
Now c and d are definite constants which will represent
A's estimate of the total mass and total momentum of the
system whatever happens to the bodies, given there are
no external forces. The bodies may collide and so cause
a transference of momentum ; the collision may break up
one or both of the bodies into several pieces, etc. In spite
of all this, taking the system as a whole, A will always
obtain the same values for c and d : these values, there
fore, correspond to some intrinsic property of the system.
Now if O accepted the principles of Newton's mechanics,
he would agree with A that the total mass was always
m 1 +m 2 =c, and he would say the total momentum was
m^u+Vj) +w 2 (w+v 2 )
=m 1 u+m 1 v 1 +m 2 u+m z v z =u{m 1 +m 2 )+m 1 v 1 +m 2 v 2
=uc\d.
108 READABLE RELATIVITY
O would therefore not agree with A's measure of the
total momentum, but he would agree that the total mo
mentum always remained the same whatever catastrophes
occurred internally in the system. In other words, if a
system is behaving so that one observer says that the
total mass and the total momentum are each remaining
constant, any other observer would agree with this state
ment, although he might disagree with the numerical
values of the constants.
But if O accepts the principles of Einstein's mechanics,
he says that the velocities of the bodies are
u+v t u+v* .
f+wi and i+^T 2 luxes P er secon <*>
and therefore the total momentum is
i+uv x "•" I+UV 2
which equals
\t+uv 1 T.+uv % r\T.+uvi r z+uvj'
Now although, whatever catastrophes occur, the values
of m 1 \m % and w 1 w 1 +w 2 w 2 remain unchanged, yet subject
to these two conditions m lt m z , v x , v 2 may vary in value
in all sorts of ways. And we see that these two conditions
are no longer sufficient to compel O's expression for the
momentum to remain unchanged, for we cannot state it
in terms of u, c, d only.
In other words, the fact that a system is behaving so
that one observer says the total mass and the total
momentum never alter is not sufficient to compel another
observer to take the same view.
This means that we must surrender the principle of
Conservation of Momentum as a property of a system,
and make it depend on the standpoint of the observer.
Such a sacrifice would rob mechanics of one of its most
MASS AND MOMENTUM 109
fundamental principles. Einstein makes this sacrifice
unnecessary by introducing a new conception of Mass.
Einstein's Definition of Mass.
Instead of saying that the mass of a body is independent
of its velocity, we shall say that if the mass of a body at
rest in A's world is m, then if the body is moving at v luxes
per second in A's world, its mass as measured by A is
m
Vx—v 2 '
It is interesting to note that, long before Einstein intro
duced this definition of mass, experimental work had
suggested that the mass of bodies moving at very high
velocities varied with the velocity. Sir J. J. Thomson's
researches on the movements of electrons had shown that
a high velocity caused an apparent increase of mass of
amount \mv 2 \ and it can be shown (see Exercise VIII.,
No; 7) that if v is small m+\mv % is a close approximation to
m
VT^v 2 '
Taking this definition of mass, it is now possible to prove
that when one observer says that both the mass and the
momentum of a system remain constant, all other observers
will agree with him. Momentum is still defined as
mass X velocity ; therefore, if a body in A's world has a
mass m when it is at rest, we see that when it is moving
at v luxes per second in A's world, its mass is
m . mv
, — ^ and its momentum is , —  2 .
vi — v 2 v I— v
Conservation of Mass and Momentum.
Using the same notation as on p. 107, A says that the
total mass is
/ „ 1 4 , — « c, say ;
no READABLE RELATIVITY
and the total momentum is
V
Whatever (internal) vicissitudes the system undergoes, A
will always compute the total mass as c and the total
momentum as d.
Let us now examine O's calculations.
O says that the velocities of the bodies are
u+v t u+v 2
■,.„; ■, =rj —  luxes per second.
The mass of the first body is therefore
Now i ( U+Vl f = ( I +^i) 2 ~(^+ t; i) 2
_ I + 2UV 1 f U 2 V X 2 — U 2 — 2UV 1 —V 1 2
(1+WVi) 2
_ i— « 8 — p 1 8 +«V _ (i— « 8 )(i— V)
(i+^j) 2 ~~ (I+Wj) 2 '
.'. the mass of the first body is
m _. /f(i« a )(rPi a )l «i(i+wi)
1 " V \ (l+ww x ) 2 J _ Vi_ w axv'i_ l ; 1 2'
.*. the total mass
_ m^i+uvj m 2 (i+uv 2 )
Vz—u 2 X Vi— v z~t~Vl— u *x Vz—v 2 a
_ i \ m x \um$) x m^ \um 2 v 2 \
=^=^ c +^}.
MASS AND MOMENTUM in
.*. although O assigns a different value to the total mass
from that assigned by A, as long as A finds that both
the total mass and the total momentum remain
constant, we see that O will also find that the total
mass remains constant.
O also says that the momentum of the first body is
m \ u\v 1
f / U J r V 1 \ 2 I+UVi
Using the results just obtained, we see that this is equal to
VI^P X Vi — Vj * x YTuvi ° r Vl_ W 2X Vl—vf
.'. the total momentum
~ VY^u? x Vi^^ 1 2+ Vi^fi x vr~^
 vi=^\ u \ Vi=^ + Vi^r w^v^ •v / iv 2 V)
.'. although O assigns a different value to the total
momentum from that assigned by A, as long as A
finds that both the total mass and total momentum
remain constant, we see that O will also find the total
momentum remains constant.
Hence, if we accept Einstein's definition of mass, we
preserve both the Principle of the Conservation of Mass and
the Principle of the Conservation of Momentum. Both
mass and momentum are intrinsic properties of the system,
existing independently of the observers, although different
observers apply different standards of measurement to
them.
The above discussion deals only with motion in the direc
tion in which A's world is moving away from O. If a body
H2 READABLE RELATIVITY
at rest in A's world has a mass m, we say that its mass
, m
becomes ^— — ^ as computed by A when its velocity is
v luxes per second in A's world, whatever the direction of
motion of the body. In order to establish the Conservation
of Mass and Momentum in the general case, it is of course
necessary to take account of the formula for the composition
of transverse velocities ; the method is the same as before,
but of course the actual algebra is modified. We leave it
as an exercise for the reader.
The term " proper mass " is applied to the massmeasure
of a body at rest in the world in which its mass is computed.
In the argument used above, m is the proper mass of the
body in A's world.
When the body is at rest in A's world, O says that its
mass is V // I _^ 2 \  ^ nas Deen pointed out on p. 109 that
if u is small this is approximately equal to m(i+£w 2 ) or
Readers acquainted with elementary mechanics will
recognise that \mu % represents what is called the " kinetic
energy " of the body— that is to say, the amount of work
it is capable of doing by virtue of its motion. The term
" potential energy " is used for the work a body can do by
virtue of its configuration— for example, a compressed
spring is said to possess potential energy. The mass of the
body, which according to O is approximately m+$mu?, is
therefore equivalent to the sum of its proper mass and its
kinetic energy.
The proper mass is therefore of the same nature as the
kinetic energy, and we may think of this proper mass m
as representing the potential energy of the body in the
world in which it is at rest.
O's measure of the mass of the body is therefore the sum
of its potential and kinetic energy ; and the fact that the
MASS AND MOMENTUM 113
mass of a body increases with its velocity is equivalent to
the statement that an increase of (kinetic) energy shows
itself by an increase in apparent mass. This leads us to
identify mass with energy, and to treat the conservation of
mass as equivalent to the conservation of energy.
Momentum and Separation.
The comparative complication of the algebra we have
used to establish the conservation of mass and momentum
seems to make the simplicity of the result all the more
surprising. Was the new definition of mass a happy guess,
or was there some argument which indicated the form it
might be expected to take in order to assume an invariant
character when passing from one world to another ?
If we replace velocity, which is displacement per unit
increase of time, by a displacement per unit increase
of separation, our new type of momentum for uniform
motion in A's world would be represented by mx
where s 2 =* 2 — x 2 . We then have _=i— =i— v 2
s t 1
Vi— v 2 or
_ VI _ irWs _ v .
Hence m x=m X 7 X 
1 mv
=mXvx
Vi — v 2 v'i — v 2
m
=7==zXV.
vi— v 2
If then m is the proper mass of a body in A's world, we
may say that the momentum of the body is
either the proper mass X the displacement per unit
increase of separation ;
or the modified mass X the displacement per unit
increase of time,
8
ii 4 READABLE RELATIVITY
In order to preserve the principle of Conservation of
Momentum, we could therefore either modify our defini
tion of Velocity or, as is done above, modify our definition
of Mass.
The Principle of Restricted Relativity.
Throughout this book we have been continually examin
ing various phenomena from the standpoints of different
observers and attempting to coordinate the results re
corded by these observers. If any law is enunciated which
summarises physical processes, it is essential that its
validity should be recognised by all observers alike. When
such a law is expressed in mathematical form, it must
retain that form when we pass from the axes of reference
adopted by one observer to those adopted by another
observer. In mathematical language, the form must be
invariant for any necessary change of axes. We saw that
the separation between two events was an invariant in
this sense. Unless a law satisfies this condition, it cannot
be true. This is Einstein's Principle of Relativity. Hither
to we have only considered worlds moving relatively to
each other with uniform velocity ; and the existence of
this limitation is indicated by referring to the subject as
the Restricted Theory of Relativity ; its principles were
enunciated by Einstein in 1905. The General Theory of
Relativity takes into account the relations between worlds
moving relatively to each other with variable velocity ;
Einstein's investigation of this theory was not published
till 1915, and owing to the War did not attract attention
in England till 1917.
His Principle of Restricted Relativity may be stated as
follows : Every law of Nature which holds good with respect
to one coordinate system (say A's world) must also hold
good for any other coordinate system (say O's world),
MASS AND MOMENTUM 115
provided that A's world and O's world are moving with
uniform velocity relatively to one another.
This is equivalent to the statement that it is impossible
to devise any experiment which will detect uniform motion
through the ether. For, if a law was valid hi one world
only, it would indicate something unique in the nature of
that world, and enable that world to be taken as a standard
of reference. And equally if uniform motion through the
ether could be detected, we should thereby possess a
unique criterion for distinguishing one special world from
all the others.
The purpose of the Theory of Relativity is to distinguish
the subjective impressions of the observer and the relations
which connect him with what he observes from the reality
of the thing observed. The property of separation occupies
a central position because it is a property of physical reality.
The name " Relativity " is to some extent misleading, as
it tends to imply that physical inquiry is of a relative nature.
What Einstein has set himself to achieve is a formulation
of physical principles which are independent of the observer.
His work may, therefore, more justly be called a Theory of
Physical Reality, or a Theory of SpaceTime Events.
EXERCISE VIII
i. A body C is moving in the world of A along AB with a
velocity £ lux per second ; O says that A's world is
moving in the direction AkB at (i) £, (ii) & lux per
second. What is C's velocity according to O ?
2. Repeat No. i, using the same data, except that C is moving
in A's world in the direction B>A.
3. A body C is moving in the world of A at £ lux per second
at right angles to AB ; O says that A's world is moving
in the direction A>B at f lux per second. What is C's
velocity according to O ?
4. A body C at rest in A's world has a mass 2 as measured by A.
If C now moves at § lux per second along AB in A's
n6 READABLE RELATIVITY
world, what is A's measure of its mass ? If O says that
A's world is moving along AB at } lux per second, what
is O's measure of its mass ?
5. If C is in motion with the data of No. 4, evaluate C's momen
tum as measured (i) by A, (ii) by O.
6. A body, whose proper mass is 5, is moving at oi lux per
second ; show that its apparent increase of mass is
approximately equal to its kinetic energy (half mass x
velocity 2 ).
7. Show that (1 +iw 2 ) 2 (i — 1/8) = 1 _f w «— Jt/«. Hence if v* is
that 1+^v 2 is a close approximation to
CHAPTER IX
GENERAL RELATIVITY
" I have made such wonderful discoveries that I am myself
lost in astonishment : out of nothing I have created a new and
another world." — John Bolyai ; a letter to his father on
nonEuclidean Geometry, dated 3rd November 1823.
In the restricted Theory of Relativity, we have considered
only a special class of observers, namely, those who are
moving relatively to the events observed with uniform
velocity. In this theory we have seen that there is no
justification for selecting any special observer as a Court
of Appeal. It now becomes necessary to proceed to a
more general inquiry. What differences arise if observers
and events move relatively to each other with variable
velocity ? Motion of this kind is of very common occur
rence ; its investigation forms what is called the general
Theory of Relativity.
Force and Acceleration.
If a stone is dropped from the top of a tower, it falls
with a velocity which increases with the time. If we
neglect airresistance, the velocity increases at a constant
rate : after 1 second its velocity is 32 feet per second, after
2 seconds its velocity is 64 feet per second, after 3 seconds
its velocity is 96 feet per second, and so on. We say that
the body is moving with uniform acceleration, and that its
measure is 32 feet per second every second. The accelera
tion of a body is the increase of its velocity per unit time.
If the velocity is diminishing, the acceleration is negative.
n8 READABLE RELATIVITY
According to Newtonian mechanics, a body is accelerated
(i.e. its velocity is changing in magnitude or direction or
in both ways) */ and only if a force is acting upon it ; and
the magnitude of the force is represented by (i) the increase
of momentum per unit time, or in other words (ii) mass
X increase of velocity per unit time, or in other words
(iii) mass x acceleration.
If a body A pushes against a body B, it is easy to picture
the nature of the force which A exerts on B. We can
think of B as being struck by a very large number of
molecules belonging to A, moving at very high speed and
thereby transferring some of their own momentum to the
molecules of B. Force exerted by what appears to be
actual contact therefore seems an intelligible operation,
and forces of this kind play a large part in ordinary life.
The seat of the chair in which I am sitting is bombarding
me with molecules, all giving me some of their momentum,
and producing in me a consciousness of force which I call
the pressure exerted by the chair. But this momentum
I am receiving is spent as rapidly as it is given. What is
the cause of the expenditure ? According to Newton, it
is a force of attraction towards the centre of the Earth
which the Earth exerts on me — the force of gravitation.
Clearly this is a force of an entirely different character ;
it does not act by means of direct contact, but appears
as an intangible influence radiating throughout space and
diminishing inversely as the square of the distance from
the source. Mention has already been made (see p. 6)
of the difficulty, which familiarity has disguised, of accept
ing an hypothesis involving " Action at a Distance." But
there is an even more remarkable property peculiar to
gravitational forces.
Arrange an experiment so that two bodies are free to
move under the gravitational attraction of the Earth,
with no other forces influencing the motion. Newton
GENERAL RELATIVITY 119
took a closed cylinder in which he placed a feather and a
guinea, and then exhausted the air from the cylinder. The
two bodies were then allowed to fall simultaneously from
the top of the cylinder. They fell side by side and struck
the bottom at the same moment. Apart from the action
of other forces, all bodies, however different in size, shape,
or mass, fall with equal accelerations under the influence
of gravity, provided they start from the same place. In
other words, the acceleration of a body due to gravity
depends solely on its position in space and is independent
of size, shape, composition, and mass of the body. The
acceleration is a function of position : near the Earth's
surface we know by experiment that the acceleration is
approximately 32 feet per second every second. Outside
the Earth, the acceleration varies inversely as the square
of the distance from the centre of the Earth. Taking the
Earth's radius as 4000 miles, it follows that 8000 miles
^2
away from the Earth's centre the acceleration is y 2 = 8 feet
per second every second, and at 12,000 miles from the
Earth's centre it is ^=^=3'6 feet per second every
second, and so on. If we limit our attention for the
moment to bodies not more than a mile or two above the
Earth's surface and within a mile or two of each other,
we may regard the gravitational acceleration as practi
cally uniform, and we shall say that the bodies are situated
in a uniform field of force.
Fields of Force.
If an observer is watching a body in motion and notes
that it is moving in a curve, he will say that there must be
some force acting on it. This is required by Newton's
Laws.
Suppose, however, some other onlooker, watching the
120 READABLE RELATIVITY
same series of events, says that the body is moving uni
formly in a straight line and therefore is not subject to
any force. Who is right ? Is it possible for two equally
honest and competent observers to differ as to whether
the path of the body is straight or curved ? Is it possible
for the statements that (i) a force acts on the body, (ii) no
force acts on the body, to be equally true ? Is it possible
to say that one observer by virtue of his circumstances is
better qualified than the other to judge what is happening ?
Suppose A lives in a large transparent airtight glass box,
say the size of Olympia. A's home is taken to a place
several thousand feet above the Earth's surface and allowed
to fall. B stands on the ground and watches A's behaviour
through a powerful telescope. For the sake of simplicity
we shall suppose there is no airresistance.
B's observations will be based on the fact that B regards
himself as at rest on the Earth's surface and regards A as
falling vertically with an acceleration of 32 feet per second
every second. Everything in A's house is behaving in
the same way — the pictures on the walls, the pipe in A's
mouth, a tennis ball which A holds in one hand and a spring
balance which he holds in the other. When A throws the
ball across the room, B says that the ball describes in space
a curve called a parabola, the curve in which any projectile
moves.
Now consider A's sensations. If A wishes to decorate
a wall of his room with a picture, he holds it up against
the wall and leaves it there ; there is no need to suspend
it from a hook ; when he takes his hand away, the picture
remains where he has put it. B, of course, says that the
wall and the picture both fall at the same rate. A takes
his pipe out of his mouth and drops it, but the pipe remains
stationary. B says that both the pipe and A fall side by
side. When A throws the ball across the room, he judges
that it moves in a straight line until it collides with some
GENERAL RELATIVITY 121
object in the room) although B says its path is curved. A
attaches a table to the springbalance and notes that its
weight is zero ; he then stands on the platform of a weigh
ingmachine and observes that his own weight is zero.
All these experiments convince A that he is at rest in a
space free of gravitational attraction, while B is equally
convinced that A is falling in a uniform field of force.
Previous discussions in this book make it easy to reconcile
the discrepancy between the opinions of A and B. Whereas
B chooses axes of reference attached to the Earth, A takes
axes of reference attached to the glass box. These two
systems are moving relatively to each other with uniform
acceleration ; and the consequence of this is that a path
A calls straight, B says is curved, and a region which A
declares is free of force, B says is a uniform field of force.
The relation is reciprocal : A says that the Earth and B
are falling towards him with a uniform acceleration. If
B rolls a ball across a level table, A says that the ball
actually describes a parabola in space, and so on.
The Principle of Equivalence.
Our natural inclination is to associate ourselves with
B's view rather than A's, but this is a parochial attitude,
and is due to the fact that we are accustomed to live under
the same conditions as B does. If, however, we were a
race of falling aviators we should sympathise with A
rather than B. The Theory of Relativity forbids us to prefer
any one observer to any other ; there is to be no favouritism
or prejudice : any law of Nature must be equally acceptable
to all observers, and must therefore take an invariant form
which survives transformation from one world to another.
Gravitational force is an illusion. This does not, of course,
mean that if you throw yourself off the top of a tower,
trouble will not ensue. But Einstein denies that the
event consisting in your subsequent collision with the
122 READABLE RELATIVITY
ground is caused by the Earth exerting an attractive force
on you. We shall soon be in a position to give Einstein's
explanation of the course of events.
So far we have only considered the effect of gravitation
over a small region throughout which the Newtonian
theory regards it as setting up a uniform field of force.
We have seen that in this case the effects can all be removed
by a change of axes. The existence of a uniform field of
force as affirmed by B is denied by A, who chooses axes
moving with uniform acceleration relative to B's.axes.
Another observer with a different system of axes would
affirm the existence of a different field of force. Conse
quently we may say that these fields of force are imputed
by the observer to the Universe owing to his own local
circumstances. A suitable change of axes will neutralise
any uniform field of force. It therefore follows that a
uniform field of force is artificial, an unconscious invention
of the observer, rather than a property of the thing observed.
A has by his choice of axes neutralised in his own neigh
bourhood what B calls a gravitational field, but in so doing
he has made B appear to be falling toward him with an
acceleration 32 feet per second every second, and if he can
see through the Earth and observe another aviator C
crashing there, he will impute to C an acceleration of 64 feet
per second every second. A therefore says that B is in a
field of force and C in another field of double the intensity :
similarly with other falling aviators elsewhere, A will
impute to each a field of force of different intensity and
direction. Consequently, although A's choice of axes re
moves the effect of gravitation in his own neighbourhood,
it makes matters worse elsewhere, by creating fields of
force of all sorts of various magnitudes and directions.
However, that does not perhaps matter to A. The con
clusion of these remarks forms Einstein's Principle of
Equivalence : 4
GENERAL RELATIVITY 123
If attention is confined to a small region of space, a gravita
tional field at rest is equivalent to a frame of reference moving
with uniform acceleration in a field free of gravitation ; and
it is impossible to devise any experiment which will distinguish
between the two.
We therefore see that although the presence of matter
is responsible for creating a gravitational field, yet any
observer, just like the inhabitant of the glass box, can so
choose his axes that in his immediate neighbourhood all
gravitational effects are neutralised, and consequently,
within this small region, although not beyond it, the
principles of the theory of restricted Relativity apply. This
conclusion is of the utmost importance, because it enables
us to use within these limits results established for space
time uninfluenced by matter.
SpaceTime Distortion.
Let us now consider the worldline of a body moving
through a spacetime domain, in which matter is present,
and let us suppose that an observer is moving with, in fact
travelling on, the body. At each point in spacetime, the
observer by his choice of axes can and does neutralise the
gravitational field in his immediate neighbourhood. He
can use his own clock to measure the separation of two
events in his career if these events are very close together ;
the separation will be simply the proper time as recorded
by his clock. By a process of summation he will then find
the total measure of the separation between any two events
in his career, measured along his worldline. If the body
is moving freely, the observer moving with it says that the
worldline is straight, and consequently that the path in
spacetime is such that the separation between two events
is a maximum. If then the measure of the separation is
the same for all observers, the path must be such that every
other observer will find that the separation measured
124 READABLE RELATIVITY
along it is a maximum. But other observers will not say
that the worldline is straight ; from their points of view the
principles of the restricted theory do not apply. The fact
that the geodesic, i.e. the route for maximum separation,
is a straight line depends on the restricted theory where
spacetime is uniform. This is no longer true.
The other observers say that the presence of matter has
distorted spacetime in its neighbourhood, and as a result
of this the geodesic is a curve. According to Newton, the
Earth describes an ellipse round the Sun owing to an
attractive force which the Sun exerts on the Earth. But
according to Einstein the presence of the Sun causes irregu
larities in the spacetime in its neighbourhood, and the
Earth simply picks its way through this tangled domain
following a path (can we call it a spiral ellipse in space
time ?) so devised that when we allow for the crumpling
up of spacetime the separation measured along it between
any two given events is a maximum. In other words, the
Earth's orbit is curved not because the Sun exerts any force
on it, but because in the distorted spacetime domain round
the Sun the geodesic is not straight but curved : it is easier
to move through the obstacles by following a curved route,
just as in passing through a wood in which the trees are
denser in some parts than others it is often easier to follow
a curved route than to try to go straight ahead all the time.
This conception of distorted space may be easier to
appreciate by considering another illustration of a field of
force.
Life on a Rotating Disc.
Imagine a large plane disc, centre C, which an outside
observer O says is rotating about an axis through C per
pendicular to its plane. Another observer A lives on the
disc and draws through C axes of reference on the disc
along and perpendicular to CA.
GENERAL RELATIVITY 125
A regards the disc as at rest, and thinks that O is moving
in a circle in the reverse direction. But A realises that he
has to attach himself to the disc
in order to keep his footing.
A believes that there is a gravi
tational field of force acting . . . ( . x
outwards from C and propor \ \^ c \J A J O
tional to the distance from C.
But O says that A is travel
ling round C in a circle with Fig. 32.
uniform speed ; and therefore
has an acceleration towards C which is produced by A
holding on to the disc, just as a stone attached to the
end of a string and whirled round in a circle is held to its
circular path by the pull of the string. Suppose now a
body starts from C and moves with uniform velocity along
CO towards O. Then of course O says that it is travelling
in a straight line in a field devoid of force. How will A
view the progress of the body ? A thinks that the disc is
at rest and that it is O who is revolving. Consequently
A will say that the body travels outwards from C along the
line CO, which is itself rotating. A therefore, tracing the
position of the body relative to his axes on the disc, says
that the body describes a sort of spiral curve ; and natur
ally he attributes the curved orbit to the gravitational
field which he believes is existing. What therefore O
regards as a straight path in a field of no force, A regards
as a curved path in a gravitational field.
Suppose now a circle is drawn on the disc with C as centre,
and that A uses his rule to measure (i) its diameter, (ii) its
circumference. Suppose that A finds the diameter is equal
to 1,000,000 lengths of his rule, O will agree with this
measurement, because in any radial position the rule has
no velocity in the direction of its length, relative to O. But
when A places the rule tangentially to the circle and
126 READABLE RELATIVITY
proceeds to measure the circumference by stepping it off
in small bits, the rule has a velocity in the direction of its
length relative to 0, and therefore says that the rule
contracts. O knows that the circumference of the circle
is equal to 7rX diameter, where ^=314159265 . . ., and
therefore 3,141,592 steps would be required to traverse the
circumference if the rule did not contract, but owing to
the contraction more steps will be necessary ; the number
of course depends on the contraction ratio. O watches
A perform the process, and notes that it takes (say) 3,300,000
steps, using the contracted rule. O and A must of course
agree in any counting process. A is surprised by this
result, because he is unconscious of any contraction of the
rule, and is forced to believe that the ratio of the circum
ference to the diameter is no longer 314159 . . ., but is in
this case 33.
A now repeats the process with a larger concentric
circle ; suppose its diameter is double that of the first
circle. O and A then agree that the diameter is 2,000,000
steps of the rule. O says that the speed of A is now twice
what it was before, and therefore the contractionratio
is greater than before, and so it now takes 8,000,000 steps
of the rule to measure round the circumference (see Exer
cise IX., No. 7). A is therefore compelled to say that the
ratio of the circumference to the diameter is ~—
2,000,000 ^
We see, then, that in A's world the circumference of a
circle is not proportional to the diameter ; in other words,
two circles of different size are not similar (i.e. are not the
same shape). A's geometry does not therefore agree with
the geometry of Euclid, and we say that A's space is
nonEuclidean.
But there is also another curious feature of A's world.
Since the velocity of A relative to O increases proportionally
to the distance of A from the centre C, O says that the
GENERAL RELATIVITY 127
clocks in A's world do not run at the same rate ; the
farther a clock is from C, the slower it runs according to
O. Timemeasure, as judged by O, is therefore not uni
form in A's world. In the restricted theory we saw that
said A failed to synchronise his clocks and that all the
clocks ran slow, but O admitted that they all ran at the
same rate : time was uniform all over A's world, although
its measure was different from that used by O. Here,
however, there is a new element of irregularity, for^A's
clocks run at a rate which depends on their distance from C.
The irregularity of spacemeasurement is also accom
panied by an irregularity of timemeasurement. The
spacetime world of A is distorted, both in respect of time
and in respect of space.
O, of course, considers that both space and time are
uniform : the nonEuclidean character of A's space and
the irregularity of time are due to A's creation of a gravita
tional field, arising from his choice of axes. O's choice of
axes has made the spacetime domain uniform ; A's choice
of axes is equivalent to setting up a gravitational field
which shows itself in the distortion of space and time.
Gravitational Fields.
The existence of matter gives rise to a gravitational
field in its neighbourhood ; but instead of saying that this
is a field of force, we now say that it is a distortion of space
time. The invariant expression for the separation between
two events, s 2 =t 2 — x 2 — y 2 — z 2 , was established by assuming
that spacetime is uniform : since in the neighbourhood
of matter this is no longer the case, this expression will
require modification in a gravitational field, if it is to sur
vive transformation from one world to another. A new
geometry is automatically imposed upon us, with a different
set of mensuration formulae. Various systems of geometry
have been investigated during the last hundred years.
128 READABLE RELATIVITY
Up till that time it had been assumed that the geometry
of Euclid was the only possible logical system. The
characteristic property of Euclid's geometry is that the
sum of the angles of a triangle is two right angles. It is
now universally accepted that equally consistent Geometries
exist which conflict with Euclid. In what is called Hyper
bolic Geometry the angle sum of every triangle is less than
two right angles, and in Elliptic Geometry the angle sum
is always greater than two right angles. If we ask which
of these is really true, the question can only mean, which
of these applies to the world in which we live. Gauss
attempted to answer the question by taking a large triangle
whose corners were the summits of three mountains and
measuring the angles. But the difference of their sum
from two right angles was less than probable experimental
errors. There is no doubt that only a triangle whose
sides involve lengths of astronomical magnitude can give
a decisive answer to an inquiry conducted in this way.
Gauss's experiment could not possibly lead to a decision.
Such evidence as exists at the present time points to the
theory that the geometry of our Universe is Elliptic, and
this involves the supposition that it is finite in extent,
finite but unbounded, just as the surface of a sphere is
finite but without a boundary.
The educated man of today understands the general
characteristics of the Newtonian theory, but only the
specialist can read and understand the Principia in which
the formal investigation is made. In the same way the
mathematical process used by Einstein to deduce the laws
of the spacetime geometry of our Universe and the formu
lation of those laws in all their generality can only be ap
preciated and apprehended by the mathematical specialist.
But the nature of the ideas which distinguish Einstein's
theory from Newton's can be illustrated without any
advanced mathematical reasoning, and it is possible to
GENERAL RELATIVITY 129
state in a simple form Einstein's law of gravitation for the
special case which affects us most, namely, for the portion
of spacetime round about the Sun.
Suppose that S is the centre of a massive body such as
the Sun, and suppose that P and Q are two events near
together in space and time. From
Q draw the perpendicular QN to SP,
produced if necessary. Since Q is
near P, we regard QN and PN as
small compared with SP, and the
length of each is to be measured
in luxes. Suppose also that the
timeinterval from P to Q is t
seconds where t is also small. Then
if m is the gravitational mass of the massive body at S,
also measured in luxes, the separation of Q from P is
given by
*(i»y_ 0N ._( I+ ») . p N *.
The method for calculating m is shown in Exercise IX.,
Nos. 5, 6. For the Sun, m =0000,004,9, and for tne Earth
m =0000,000,000,02. It should be noted that if m=o, this
expression becomes s 2 =t 2 — QN 2 — PN 2 =* 2 — PQ 2 , which is
the ordinary form for the separation in the restricted theory :
putting m equal to zero is of course equivalent to saying
that no matter is present to influence the nature of the
spacetime in which the events occur. The introduction
of the additional terms involving m therefore represent
the changes in the mensuration formulae caused by the
distortion of the spacetime domain surrounding a single
massive body.
If a body is moving freely in the neighbourhood of a
massive body, it follows a path so chosen that the separa
tion, as detennined by the formula given above and
measured along this path, is a maximum. Newton's Law
9
130 READABLE RELATIVITY
of Gravitation is replaced by Einstein's geometrical men
suration formula for a spacetime domain, a geometrical
spacetime Law. Einstein's Law may be tested by
examining the paths in which the planets move round the
Sun. Are these paths geodesies in spacetime, when the
separation is calculated according to the formula given
above ? That the paths of the planets fit very closely
with the paths as calculated in the Newtonian theory is
well known. It is simply because Newton's Law of
Gravitation leads to orbits which agree so closely with the
observed orbits that up till the time of Einstein this law
was universally accepted. But we shall see in the next
chapter that close as is the agreement between calculation
and observation, a still higher degree of accuracy is secured
by the substitution of Einstein's Geometrical Law for
Newton's Mechanical Law : the latter may indeed be
regarded as a first approximation towards the former.
EXERCISE IX
i. A toy pistol is pointed straight at the bull'seye of a target.
At the moment the pellet leaves the pistol, the target is
allowed to fall vertically. Will the pellet strike the
bull'seye, if air resistance is ignored ? Compare the
views of the path traced out by the pellet formed by
(i) the boy who holds the pistol, (ii) a microbe on the
bull'seye.
2. Suppose that A, whose weight is 10 stone when on the
ground, is standing on a weighingmachine in his glass
box (p. 120), and notices that his weight has changed
from zero (i) to 10 stone, (ii) to 20 stone, (iii) to 100 stone,
what will he say about gravitation ? How will B, who
is standing on the ground, account for it ?
3. Three people are watching a body. One says it is at rest,
the second says it is moving in a straight line, and the
third says it is moving in a curve. Is it possible that
all three observers are equally efficient ?
4. Suppose that the disc on p. 125 is rotating at the rate of
GENERAL RELATIVITY 131
5 revolutions per minute, and that a body is moving at
a uniform rate of 5 feet per minute just above the disc
along the straight line CO. Plot its positions at intervals
of 1 second for 12 seconds, as recorded by A, who uses
lines marked on the disc as axes of reference.
If the gravitational mass of the Sun is m, the acceleration
towards the Sun of a planet at a distance of r luxes from
the centre of the Sun is approximately ~^. If the planet
is moving in its orbit at v luxes per second, it is known
that the acceleration radially inwards is — . Hence
fyt v^
a=— or m=v*r. Taking the distance of the Earth from
the Sun as 500 luxes, use this formula to show that the
gravitational mass of the Sun is about 15 km., and
express the result in luxes.
Taking the distance of the Moon from the Earth as 240,000
miles and the period of a revolution as zj\ days, use the
formula in No. 5 to show that the gravitational mass of
the Earth is about 5 millimetres, and express the result
also in luxes. [1 lux = 186,000 miles.]
In the measurement of the circumferences of the two circles
on the rotating disc on p. 1 26, show according to O that
(i) the first speed of A is given by i — u a = (^^J , (ii) the
contractionratio along the larger circle is about 079,
(iii) the number of steps of the rule round the larger
circle is about 7,970,000.
CHAPTER X
THE EINSTEIN TESTS
" In one sense deductive theory is the enemy of experi
mental physics. The latter is always striving to settle by
crucial tests the nature of the fundamental things : the former
strives to minimise the successes obtained by showing how
wide a nature of things is compatible with all experimental
results. — A. S. Eddington, Mathematical Theory of Relativity.
A scientific theory maintains its position only so long as
it harmonises observed facts. If discrepancies between
theory and observation remain, after full allowance has
been made for possible observational errors, then modifica
tions must be made in the theory.
The Perihelion of Mercury.
For a long time it had been realised that there was a
serious difference between the observed orbit of Mercury
and the path obtained by calculations based on Newton's
Law of Gravitation. If Mercury were the only planet in
the solar system, its path would be an oval curve called
an ellipse having the Sun at a point inside it known as the
focus S ; the centre C of the curve
is a different point ; if CS cuts the /^ ~^^
orbit at A, A' as shown, then the Ar— 1+ JA
planet is nearest the Sun when at \ y
A and is farthest from the Sun when p IG 34
at A'. The point A is called the
■perihelion of the orbit. Owing to the attraction exerted by
the other planets, using Newtonian language, the regularity
134 READABLE RELATIVITY
of motion is upset, and instead of moving in exactly the same
elliptic orbit, year after year, the path is represented by
an ellipse in which the perihelion is steadily advancing, i.e.
SA is steadily rotating relatively to the fixed stars. Calcula
tions reposing on Newton's Law of Gravitation show that
the combined influence of all the known planets would
cause a rotation of 532 seconds of angle per century.
Observation, however, shows that it is actually rotating
at the rate of 574 seconds per century. There is con
sequently a discrepancy of 42 seconds per century to be
accounted for. This may sound a very small error — it is
less than the angle which a halfpenny subtends at the eye
from a distance of 135 yards— but actually it is far greater
than any possible observational error.
The irregularities in the motion of Uranus were respon
sible for the dramatic discovery of Neptune. In 1846,
Adams and Le Verrier, working independently, calculated
the path of a planet which would produce the observed
perturbations in the orbit of Uranus. The calculated
position in the sky of this hypothetical planet was sent by
Le Verrier to Dr. Galle of Berlin, who at once turned his
telescope to the place indicated in the sky and discovered
the new planet Neptune very close to the position predicted
for it. An attempt was made to account for the irregularity
in Mercury's motion in a similar fashion : and a planet
to which the name Vulcan was given was invented for the
purpose. But Vulcan has never been found, and its
existence is now discredited. If, however, we substitute
Einstein's Law of Spacetime for Newton's Law of Gravita
tion, the discrepancy disappears. The deduction of this
result from Einstein's Law requires mathematics of too
advanced a character for these pages, but the additional
correction which Einstein's Law supplies can be stated in
simple language : if a planet describes its orbit with a speed
of v luxes per second, the line joining the Sun to the
THE EINSTEIN TESTS 135
perihelion rotates through an additional angle of amount
I2v 2 right angles per revolution. The reader can easily
verify (see Exercise X., No. 3) that in the case of Mercury
this correction amounts to within a second of the needed
42 seconds per century.
It is very unfortunate that it is impossible to check this
correction by reference to any of the other planets. For
in the case of every other planet either the speed is too
small or else the orbit is so nearly circular that accurate
observation of the position of the perihelion is impossible.
But the fact that it gives so accurately the necessary
correction for Mercury's motion is a powerful and striking
argument in favour of the Relativity theory.
In addition to showing how his theory removed an
anomaly of which no satisfactory explanation had pre
viously been given, Einstein made two predictions which,
if capable of being submitted to an experimental test,
would serve to distinguish between the old theory and
the new.
Shift of Spectral Lines.
The vibration of an atom may be regarded as supplying
us with an ideal natural clock. If two atoms are identical,
and if we measure the separation between the beginning
and end of a vibration, the result should be the same
wherever the atoms are situated. Now suppose one of
the atoms is close to the surface of the Sun and that the
other is in a laboratory on the Earth. We may regard
the events for each atom as happening at the same
places.
Suppose that the period of vibration of the solar atom
is t x seconds, and for the terrestrial atom is t 2 seconds,
/ 2W\
then, for the solar atom, s a =^i — gp^i
136 READABLE RELATIVITY
where SP=43o,ooo miles=23 luxes, ^=0000,004,9 luxes.
And for the terrestrial atom, s 2 =(i^V 2 2
by 7
where SQ =93,000,000 miles =500 luxes,
m =0000,004,9 hixes.
The value of s is the same in each case
• • v I— i^Ai =V I— 5o^M where m =0000,004,9.
It is, therefore, clear that t x is a little greater than t 2 .
The reader may see for himself that j is approximately
1000,002. Consequently the solar atom vibrates just a
little more slowly than the atom on the Earth. Now the
time of vibration affects the colour, and therefore in the
solar spectrum there should be a very small shift towards
the red end of the spectrum as compared with the spectrum
of the same atom on the Earth.
The shift is, however, in general so minute as to defy
measurement. But Professor Eddington pointed out some
time ago a case where a larger shift might be expected.
There is a companion star of Sirius, known as a " white
dwarf," i.e. an early type of star of very low intrinsic
brightness. It is believed that the density of this star
(the statement sounds incredible !) is more than 30,000
times the density of water — that is to say, its mass is more
than half a ton per cubic inch ; its radius is about 12,000
miles. Consequently it sets up a gravitational field of so
great an intensity in its neighbourhood that a measurable
shift is likely to occur. Mr. Adams of Mount Wilson
Observatory has recently obtained results for the displace
ment of some of the hydrogen lines of the spectrum of
this star, which appear to confirm the Einstein prediction,
although the method involved considerable difficulties
both of observation and measurement.
THE EINSTEIN TESTS 137
Curvature of LightRays.
Newton envisaged the possibility that light has weight.
It is now an accepted fact that a ray of light does exert
radiation pressure on any object on which it impinges :
this is equivalent to saying that light possesses mass.
Naturally the quantities involved are small : Professor
Eddington, in his volume Space, Time, and Gravitation,
states that the mass of the total amount of sunlight im
pinging on the Earth every twentyfour hours is about
160 tons. But if light has mass, whether we follow Newton's
or Einstein's Law, a ray of light passing near the Sun
should move in a curve in just the same way as do the
planets or comets. The fact that light moves so much
faster than any planet or comet will naturally mean that
the amount of the deflection when passing near the Sun
is very much less.
If, then, a ray of light travel
ling from a star P towards the „*' _
Earth E passes near the Sun S
and is slightly deflected so that
ES is not in the same straight
line with SP, then the star,
as seen from the Earth, will Fig. 35.
appear to be in the direction
ESQ, whereas its true direction is the line EP. The
angular displacement between the true position and the
apparent position of the star in the sky is represented
by the angle PEQ. This angular displacement can be
calculated, but different results are obtained according as
we adopt Newton's or Einstein's Law. It has already been
pointed out that Newton's Law may be regarded as a
first approximation to Einstein's Law. The latter, so to
speak, adds on to Newton's Law a correction arising from
the] distortion of space in the neighbourhood of matter.
138 READABLE RELATIVITY
If we were to replace the formula on p. 129 by the relation
i~Sp,FQN 2 PN 2 , we should obtain orbits which
agree closely with those deduced from Newtonian principles.
The addition of the term — gp . PN 2 corresponds to a
substitution of nonEuclidean for Euclidean space (not
spacetime), and it is the presence of this term which makes
a decisive difference between the calculated value of the
apparent displacement of a star in the two theories. It is
impossible to reproduce the necessary calculations in these
pages, but the result can be stated in a simple form. If a
ray of light from a star passes the Sun at a distance r luxes
from the centre of the Sun, then, as viewed from the Earth,
the angular displacement according to the Einstein theory
is — right angles, where as before m =0000,004, 9 and
^=314. If the ray passes close to the Sun's surface we
may take 7=700,000 km. =23 luxes. We leave it to the
reader (Exercise X., No. 4) to show that this is equivalent
to an angle of about 175 seconds. The Newtonian theory,
on the other hand, leads to a displacement of — right
angles, just half the amount required by Einstein. It
lies with the astronomer by actual observation to judge
which is correct.
Sir Oliver Lodge, in the course of an article in the Nine
teenth Century, gave a vivid illustration of the task this
test imposes on the practical astronomer ; part of it we
venture to quote :
" Take a fine silk thread of indefinite length and stretch
it straight over the surface of a smooth table. Imagine a
star at one end of the thread and an eye at the other end,
and let the thread typify one of the rays of light emitted
from the star. Now take a halfpenny and place it on the
table close to the thread so that the eye end of the thread
THE EINSTEIN TESTS 139
is 10 feet away ; and then push the halfpenny gently
forward till it has displaced the thread the barely per
ceptible amount of T ^ inch. The eye, looking along the
thread, will now see that the ray is no longer absolutely
straight ; in other words, the star whose apparent position
is determined by that ray will appear slightly shifted. The
scale is fixed by the size of the halfpenny, whose diameter,
1 inch, is used to represent the Sun's diameter of 800,000
miles. The 10foot distance between eye and Sun practi
cally supposes the eye is on the Earth, which would be a
spot about the size of this full stop. As for the distance
of the star at the far end of the thread, that does not
matter in the least ; but on the same scale for one of the
nearest stars the thread would have to be about a thousand
miles long. The shift of ^Vjt inch at a distance of 10 feet
corresponds to an angle of if seconds, which is just the
optical shift that ought to occur, according to Einstein,
when a ray from the star nearly grazes the Sun's limb on
its way to a telescope."
Modern apparatus and methods have now attained
such a high degree of refinement that the measurement
of even so small an angular displacement as is indicated
by this illustration, or rather the discrimination between
two small angular displacements of this order of magnitude,
is well within the powers of the presentday astronomer.
Unfortunately the only time when a star nearly in line with
the Sun is visible is during a total solar eclipse, and, in
addition to this, clear observations are difficult to obtain
unless there happen to be at the time of the eclipse several
bright stars in the direction under survey. By good
fortune, a total solar eclipse occurred on May 29, 1919,
when these conditions were satisfied. Two expeditions
were organised, one being sent to Sobral, in North Brazil,
and the other to Principe, in the Gulf of Guinea, to take
the necessary photographs. The story of these expedi
140 READABLE RELATIVITY
tions has been told in detail by Professor Eddington (see
Space, Time, and Gravitation, ch. vii.) ; the expedition to
Principe fared badly, because clouds interfered seriously
with the operations, but at Sobral the atmospheric condi
tions were excellent. Here, however, there were other
complications which diminished the value of many of
the photographs. Allowing for probable experimental
errors, the Principe observations gave an apparent dis
placement of between 191 and 131 seconds, while the Sobral
observations gave a displacement between 2 10 and 186
seconds. Another total eclipse occurred in 1922, and a
successful expedition was organised by the Lick Observa
tory. The published results of one set of photographs
{Lick Observatory Bulletin, No. 346) give the mean value
as 172 seconds (or if a certain correction is applied as
205 seconds) with a possible error in defect or excess of
about 012 seconds. It is believed that the records of
another set of photographs, not yet published, are even
more favourable to the Einstein deflection. There will
be another very favourable solar eclipse in 1938.
Many readers may feel disappointed that there should
appear to be so wide a margin in the recorded results.
It is necessary to engage in practical work to appreciate
that sources of experimental error are inevitable : all that
can be done is to indicate the probable margin of error.
It is also necessary to remember that the conditions under
which an expedition has to work are far less favourable
than those of a permanent observatory, just as a field
telephone company on active service is far less favourably
placed than a London Telephone Exchange. But the
records undoubtedly support Einstein as opposed to
Newton. If it is necessary to choose between the two,
there can be no doubt that the Einstein calculations are
in closer agreement with eclipse observations than those
based on the Newtonian theory. But apart from the con
THE EINSTEIN TESTS 141
crete evidence furnished by observation, it is important to
remember that the deflection had not been suspected until
Einstein predicted it as a consequence of Relativity. A
theory which predicts a hitherto unknown phenomenon
afterwards verified by observation stands on firmer ground
than a theory invented to account for some known observa
tional effect, mainly because there is usually a variety of
different hypotheses which can be suggested to fit a given
frame of facts. And whatever modifications future re
search may require, nothing can obscure the dramatic
character of the success which has attended this prediction,
made by Einstein in 1915, and tested by astronomers at
the eclipses of 1919 and 1922, and now generally accepted
as fulfilled.
General Conclusions.
The Restricted Theory of Relativity may be regarded as
complete : future investigations will add nothing to it. If
we accept the two fundamental axioms on which it is based,
the conclusions follow as a matter of formal logic. In
process of time the characteristic ideas of the theory will
become familiar to the man in the street, their acceptance
will then become a matter of course, and their revolutionary
nature will be forgotten. If intercourse should ever become
practicable between beings in worlds separating with
velocities comparable with that of light, the theory would
seriously affect conduct, but this contingency seems so
remote and improbable that it may safely be discounted.
The General Theory of Relativity may in a sense be said
to be still in the making. Certainly its implications are
far more controversial. To say that Einstein's Law of
Spacetime has superseded Newton's Law of Gravitation
is not the same as affirming that Einstein's theory fits in
with all the phenomena of modern physics. No link has,
for example, yet been forged between the theory of Rela
142 READABLE RELATIVITY
tivity and the Quantum theory. But if we accept the
principles of Restricted Relativity, Newton's Law of Gravi
tation cannot stand in its existing form for the simple
reason that it is ambiguous. Apart from the doubt in the
Newtonian expression —$ as to the meaning of mass,
which arises because the mass changes with the velocity,
we have seen that the value of r also depends on the
circumstances of the observer.
The universality of character which was the most striking
feature of the law has therefore disappeared. No doubt
the statement by which the law is expressed could be
modified so as to remove these difficulties of interpretation.
It is not, however, worth while attempting to do so, because
it cannot in any case be expressed in a form which will be
true for all observers. It is not therefore the kind of law
which our previous discussions have shown a law of Nature
should be ; at the same time calculations based on it give
results which agree closely with those deduced from
Einstein's general theory. Formally the two theories have
nothing in common ; they are built upon contrary hypo
theses. Newton assumes an absolute space and an
absolute time, and his laws of motion are bound up with
these hypotheses. Einstein treats these suppositions as
untrue, and creates a spacetime (not a space and time)
domain. The great achievement in his theory is the fact
that he is able independently of axes of reference to identify
a unique track in spacetime uniting two events, the
generalisation of a straight line in Euclid, and to specify
this path both in a field devoid of matter and in a field
influenced by matter. The theory of Relativity, whether
restricted or general, is a theory of geodesies. The treat
ment in this book is confined to those aspects of the subject
which may reasonably be regarded as established or at least
supported by strong evidence, although further investiga
THE EINSTEIN TESTS 143
tions may introduce minor modifications. But the theory
has led physicists to formulate further hypotheses which at
present must be considered mainly speculative. What is
the nature of the structure of the Universe ; is it finite or
infinite, is it continuous or discrete, is its substance matter
or events? Some, perhaps all, of these questions may
never receive a final and complete answer. In regard to
the first of these inquiries there is, however, a slight balance
of evidence in favour of supposing that the Universe is
finite, but of course unbounded, and an estimate of its size
is expressed by the statement that a ray of light emitted
from a source would, if unimpeded, travel round the
Universe and return to its point of departure after a
thousand million years. It is not inconceivable that science
may at some future time be able to devise and execute a
practical test which will decide the question. But this is
a superficial type of knowledge. The theory of Relativity
may describe the laws which reality obeys, and trace its
structure : this indeed is all it sets out to do. Of the inner
nature of things, it says nothing : this is left to the phil
osopher, who says a great deal, but in the end we never
appear to be much, if any, the better for it.
EXERCISE X
1. At what distance would a halfpenny (diameter i inch)
subtend an angle of i second ?
2. Taking the mean distance of Mercury from the Sun as
37,000,000 miles, and the length of Mercury's year as
88 of our days, show that the speed of Mercury in
her orbit is slightly less than ^^ lux per second,
[i lux=o= 186,000 miles.]
3. It can be deduced from Einstein's spacetime law that if
the speed of a planet is v luxes per second, the major
axis of its orbit advances i2i> 2 right angles per revolution.
Using the data of No. 2, show that in 100 of our years
the major axis of Mercury advances about 43 seconds.
144 READABLE RELATIVITY
4. With the data on p. 138, show that — right angles is
approximately 175 seconds.
5. Taking the radius of Jupiter as 43,000 miles, and its mass
as jjfe^ of that of the Sun, show that, according to the
Einstein theory, the apparent displacement of a star so
placed that the light ray to the Earth just touches
Jupiter's surface is about 0017 second.
6. With the data on p. 136 for the spectral shift, prove that
J^=I000002.
h
ANSWERS
EXERCISE I
3. 185. 6. ioo seconds. 8. 17 miles. 9. 39,800 stadia.
EXERCISE IIa
1. 4 feet. 2. 2 feet ; 6 inches.
3. 60 feet ; 3 inches ; 3 inches ; 4.
4. 26 inches; 6 inches. 5. 025 ; i=(£) 2 .
6. Frustrum of pyramid, approx. £ by J by i inch.
7. 3 inches broad ; f inch thick. 8. Too thin.
EXERCISE IIb
3. (i) Ordinary ; (ii) dial oval ; hands expand and contract
as they rotate.
4. As in 3 (ii). 5. 6 3 £° N. of E. 6. No.
EXERCISE III
1. 10 yards ; 24 yards. 2.  y feet ; £ z feet.
3. 9 seconds ; 27 feet. 4. 20 seconds ; 25 seconds.
5. 32 feet. 6. Bullet ; together ; noise.
EXERCISE IV
1. (i) P 20, A 16, C 10 ; (ii) P 25, A 20, C 14 seconds past zero.
2. (i) P 30, A 24, C 1 5 ; (ii) P 37i, A 30, C 21 seconds past zero.
3. (i) D 12 ahead of A ; (ii) 15.
4. (i) P 5, A 4, C 10 ; (ii) P 25, A 20, C 26 seconds past zero.
5. (i) Event at A ; (ii) event at C. 6. Yes. 7. 40 legs.
EXERCISE V
1. $ foot when pointing east ; loses 12 minutespaces per
hour ; A says the same about O.
2. A is 1*4 secondspaces ahead of C ; D is 28 secondspaces
ahead of A ; A 24, C 226, D 268 seconds past zero.
3. 8£ seconds ; "j\ luxes. 4. 4 seconds ; same place.
10
146 READABLE RELATIVITY
EXERCISE VI
2. Yes; £ ; 113. 3. Xwm ^ t /=>6 . 4
4. 9 luxes ; 12 ; / 5 or £ lux per second.
5. 4 seconds, 7 luxes ; \ second ; A says II. occurs before I. ;
^33
6. Imaginary ; indeterminate. 7. Real ; fixed. 8. Yes.
9. III. occurred after I. and II. ; no timeorder exists for
I. and II.
10. 25 hours ; o.
EXERCISE VII
I' 8 4 2. i6, 38, 104; (25, 5, 14 ; 775).
3. 84 seconds ; parallel to timeaxis ; proper time.
5. 160. 6. 16; 8+ V 28 = 1329; no.
EXERCISE VIII
!• § , II lux per'second. 2. o, ^ lux per second.
3. § lux per second perp. toAB; f lux per second along AB
*• 2 *> 375. 5. ii, 317.
EXERCISE IX
1. Yes ; a parabola ; a straight line.
2. B says the glass box first stops and then moves with
upward increasing vertical acceleration.
3. Yes. 5. 0000,005. 6. 0000,000,000,017.
EXERCISE X
1. 326 miles.
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